Optimal allocation patterns and optimal seed mass of a perennial plant

Optima l allo catio n patterns and optim al seed mass of a p er ennial plant Andrii Mironchenk o a, ∗ , Jan Koz lowski b a Institute of Mathematics, University of W¨ urzbur g, Emil-Fischer Str aße 40, 97 074 W ¨ urzbur g, Germany b Institute of Envir onmental Scienc es, Jagiel lonian University , Gr onostajowa 7, 30-387 Kr ak´ ow, Poland Abstract W e presen t a no vel optimal allo cation mo del for perennial plants, in which a ssimilates are not alloca ted directly to vegetativ e or reproductive parts but ins tead go first to a storage compartment from where they are then optimally redistributed. W e do no t restrict co nsiderations purely to p erio ds fav ourable for photo synthesis, as it was done in published mo dels of p erennial sp ecies, but ana lyse the whole life p erio d of a p erennial plant. As a result, we obta in the general scheme of p ere nnial plant dev elopment, for which annual and mono c arpic stra tegies are sp ecial cases. W e not only r e-derive predictions fr om several previo us optimal allo catio n mo dels , but als o obtain more infor mation ab out plan ts’ strategies during transitions be t ween fav ourable and unfav ourable sea sons. O ne o f the mo del’s predictio ns is that a plant can b egin to re-establish v egetative tissues from stor age, some time b efore the b eginning o f fav ourable conditions, which in turn allows for better pro duction po ten tial when conditions be come b etter. By means of numerical examples we show that annual plants with single or multiple repr o duction p erio ds, mono ca rps, evergreen p erennials and po lycarpic p erennials can be studied succe ssfully with the help of our unified mo del. Finally , we build a bridge b etw een optimal allo c ation mo dels and mo dels descr ibing trade-offs b etw een size and the nu mber o f seeds: a mo delled pla n t can control the distribution of not o nly a llo cated carb ohydrates but a lso seed size . W e provide sufficient co nditions for the optimality of pro ducing the sma llest and la rgest seeds p ossible. Keywor ds: Optimal pheno logy; size-num ber trade-off; biomass partitioning; p erennial plants 1. In tro duction The pioneering work [1] gav e rise to a new class of math- ematical mo dels of plants based on metho ds of optimal control theory . In these mo dels it was a ssumed that a plant can c ontrol resource allo c ation in order to maximise its fitness, which is often identified with the mass o f seeds pro duced by a pla n t during its lifetime. In the firs t mo dels , which were devoted to the develop- men t of annual plants, it was a ssumed tha t a plant con- sists of a num b er of compartments – at least of a vegeta- tive compartment (leaves, ro ots, stems) and a repro duc- tive compar tmen t (seeds and auxiliary tissues), although storage and defensive tissues could also b e included. This bas ic mo del, p osited by [2], results in a bang- bang transition from the allo cation to vegetative tissues to the allo cation to seeds. This annual plant mo del ha s b een ex - tended in man y directions, in particular in [3 ] a model with m ultiple vegetativ e compartmen ts w as analysed, and in [4] and [5], in which additiona l physiologica l constr aints were considered, resulting in per io ds of mixed g rowth (wher e bo th the v egetative and repro ductive par ts of a p lant grow ∗ Corresp onding author Email addr esses: andrii.m ironchenko @mathematik.uni-wuerzburg.de (Andri i Mironchen ko) , jan.kozlo wski@uj.ed u.pl (Jan Koz lowski) simult aneously). Optimal a llo cation stra tegies in sto chas- tic en vironments hav e b een inv estigated in par ticular in [6], while allo cation to defensive tissues was encountered in [7] and [8], to cite a few ex amples. The sur vey of ear ly works in this field is provided in [9 ], and for a gener al overview o f resource allo c ation in pla n ts see b o ok s [10] and [11]. In c ont rast to a nn ual pla nt s, less attention has b een de- voted to the mo delling of p erennials’ o ptimal phenology . Usually , the b ehaviour of a p erennial plant is mo delled in the following way: its lifetime is divided into discrete sea - sons du ring whic h e n vironmental conditions are favourable for photosynthesis. The mo del of a plant within every se a- son is co n tinuous a nd is treated with the metho ds used in annual plant mo dels [12], [13], [14]. T o mo del the be- haviour of a plant b etw een seas ons (when the w eather is unfav o urable), so me s imple tra nsition rules a re used that show whic h parts o f compa rtments are sav ed during the season and which are no t. The solution to such pr oblems is divided int o tw o parts: first, the mo del o n one season is solved us ing Pontry agin’s Maximum Principle (see e .g. [15]), and then a solution to the who le mo del is sought by applying the dy namic pro gramming metho d. Although these mo dels pr ovide quite interesting quali- tative results reg arding the b ehaviour of p erennial plant s, they ha ve an impor tant disadv a n tage, namely that the subtle qualita tive b ehaviour of a plant during a sea son Pr eprint submitte d to Journal of The or etica l Biolo gy Octob er 13, 2018 contrasts with the simple jump fr om the end o f one season to the b eginning o f the next one . In this pap er we prop ose a p erennial pla n t contin uous-time mo del, which allows us to des crib e more precisely the dyna mics of a pla n t dur- ing se asons with unfa vourable environmental conditions for photosynthesis, and to avoid the intro duction of addi- tional parameter s for describing jumps b et ween seasons. With the help of Pon tryagin’s Maximum Pr inciple we derive a g eneral scheme of p erennial plant developmen t, which contains mo dels for annual as well a s mono carpic plants as sp ecia l cases. W e a lso pr ov e that mo no carpy is alwa ys optimal if there are no losses o f sto rage parts a nd there is no mortality b efor e the e nd of life. With the help o f numerical e xamples we show that m any developmen tal patterns from pre vious pap er s can be de- rived by using our mo del. In par ticular, one can use it to study annual plan ts with multiple r epro duction p erio ds [18]; p erennial plants which grow to a cer tain s ize for a nu mber of p er io ds, and in the f ollowing p erio ds they fir stly regrow to this size and then pro duce repr o ductive tissues [13]; the evergreen polycar pic plan ts as w ell as mono car ps. Moreov er, o ur mo del is m uch b etter suited for the study of transitions from fa vourable to unfav ourable climate con- ditions, and one of its pr edictions is that plants be gin to generate vegetative tissues not a t a time when environ- men tal conditions ar e fav ourable for photosynthesis, but slightly earlier in o rder to enter into the suitable p erio d with developed vegetativ e tissues. Having established an optimal allo catio n mo del we will connect it to the theor y o n tra de-offs b etw e en size and nu mber of seeds. A lot of a tten tion ha s been devoted to these tr ade-offs in the s cientifi c litera ture. The basic mo del has b een prop osed in a seminal work [16], where it was assumed that the fitness of a plant is equal to the sum of the fitnesses of its desc endants. Afterwards this mo del has b een gener alised in a num ber of directions (for a r e- view see [1 7]). In this framework, optimal size is sought depe nding o n the prop erties of the fitness function. This makes po ssible the quite general treatment of s ize-num b er trade-offs, but the question remains a s to how to formalise the dep endency of fitness on size and the num be r of seeds and how to find the prop erties of the function that char- acterises this dep endency . Our aim is to investigate the trade-o ffs betw een the nu mber and s ize of seeds in the co nt ext o f optimal al- lo cation mo dels. Within this fra mew ork fitness is pro p- erly formalised, and we can investigate the optimal size of a seed dep ending on the pro per ties of the photosyn- thetic r ate function and other physiologica l par ameters of a plant, which offer more distinct criteria than abstract fitness. W e provide the analysis for the mo del developed in Sec tion 2 o f this pa per , but the results are v alid also fo r a num b er of other optimal allo cation mo dels. W e pr ov e that, a ccording to our plant mo del, if the pho tosynthetic rate function is concav e (that is, if the rate of photos yn- thesis p er unit mas s decays with an increase in the size of a plant), then the seeds have to be as small a s po ssible. Such behaviour is particula rly t ypical in co lonising sp ecies (see Section 5 ). Our mo del also includes the p ossibility of choos ing the germination time of a seed. As a co nsequence, we obtain results concerning the behaviour of plants fro m dormancy of seeds up to senile stag e. The outline of the ar ticle is as follows. In Section 2.1 we intro duce a p erennia l plant model. In Section 2.2 we summarise the predictions of the mode l, provide a general plant development scheme and co nsider a num ber of sp e- cial cases (annual and mo no carpic pla n ts). In Section 3 we make numeric simulations of different plant develop- men t scena rios, while in Section 4 we co nsider trade-offs betw een size a nd the num b er of seeds. The res ults of the pap er a re discussed in Section 5, a nd Section 6 draws con- clusions from the results and outlines some direc tions for future w ork. In Appendices 1 and 2 we provide deriv a tions of our theor etical r esults. 2. Optimal al lo cation mo del 2.1. Mo del description In optimal allo ca tion mo dels it is usually assumed that all allo cated photosynt hate is immediately used fo r the construction o f tissues. In mode ls taking int o account the presence o f a sto rage compartment, a plant can a lso al- lo cate re sources from sto rage, dep ending on the mas s of the storag e. Such a metho d ignor es the fact that a photo- synthate is not immediately allocated to certain structures but instead exis ts fo r some time in a fre e s tate. W e shall take this effect into acco un t and ass ume tha t an inter- mediate stage e xists whereby car bo h ydrates have a lready bee n photosynthesised but hav e not yet been p ermanently allo cated to a g iven structur e. Let a plant consist of three parts: a vegetativ e compart- men t, a repro ductive compartment and non-structur al car- bo hydrates (free gluco se, s tarch, etc.), her einafter called ‘storage ’. Let • x 1 ( t ) b e the mass of the vegetative compar tmen t a t time t , • x 2 ( t ) b e the mass o f the repro ductiv e c ompartment at time t , • x 3 ( t ) b e the mass of storag e at time t . W e mo del the dynamics of a plant v ia the following equations: ˙ x 1 = v 1 ( t ) g ( x 3 ) − µ ( t ) x 1 , ˙ x 2 = ( v ( t ) − v 1 ( t )) g ( x 3 ) , ˙ x 3 = ζ ( t ) f ( x 1 ) − v ( t ) g ( x 3 ) − ω ( t ) x 3 . (1) Here f ( x 1 ) describ es the r ate of photosynthesis of the plant with vegetative mass x 1 in o ptimal environmental conditions, and g ( x 3 ) - the maximal r ate of allo cation o f non-structura l carbo h ydrates. It is na tural to assume that f and g incre ase monotonica lly a nd f (0) = g (0) = 0 . 2 Climate influence is mo delled by three functions: ζ : [0 , T ] → [0 , 1] and µ, ω : [0 , T ] → [0 , ∞ ), where T is max i- m um longevity . • ζ ( t ) models the dependence of the ra te of pho tosyn- thesis on the c limate ( ζ ( t ) = 0 if at time t no photo- synthesis is p ossible); • µ ( t ) is the loss ra te of v egetative tiss ues per unit mas s at time t ; • ω ( t ) is the loss r ate of the stora ge parts p er unit mass due to ex ternal factors (decaying, gr azing by animals, etc.) at time t . Note that photosynthesised car bo h ydrates fir stly e n- large the mas s of stora ge b efore they can b e a llo cated to other compartments. W e assume that a pla n t can control the total allo catio n rate with the control v ( t ) ∈ [0 , 1] and the allocation rate to vegetativ e tissues with the co nt rol v 1 ( t ) ∈ [0 , v ( t )]; conse- quently , the allo ca tion rate to r epro ductive tissues at time t is controlled b y v 2 ( t ) = v ( t ) − v 1 ( t ), a nd v ( t ) = 0 means that resources are n ot being relo cated from s torage at time t . The initial ma ss of the seed and all its compartments is given as follows x i (0) = x 0 i , i = 1 , 2 , 3 . (2) Problem o f optimisation of a seed mass will b e considered in Section 4. Seed do rmancy is modelle d as the ability o f a plant to choose the time of germinatio n t 0 ∈ [0 , T ]. F or simplicity we assume that a seed cannot decay and it do es not use a ny resource s fo r life-sustaining activities b efore germina tion. Thu s x i ( t 0 ) = x i (0) = x 0 i , i = 1 , 2 , 3 . (3) T o mo del the mortalit y of a parental plant, w e in tro duce the function ˜ L : [0 , T ] → [0 , 1]. ˜ L ( t ) is the pro bability of surviv a l of a pare n tal plant to age t . W e assume in this pap er that mortality is only a ge-dep endent and doe s not depe nd on the s ize o f a plant. Since the time o f germi- nation may v ary , it ma kes se nse to introduce the function L t 0 , defined b y relation L t 0 ( t ) = ˜ L ( t 0 − t ). In what follo ws, we write for s hort L = L t 0 . It is na tural to assume that L is a non-increa sing func- tion and that L ( t ) > 0 for all t ∈ [0 , T ). In fact, if L ( t ) ≡ 0 on [ T − ε, T ] for s ome ε > 0, then this means that at mo- men t T − ε a plant will already b e dead, and we can there- fore consider the optimal con trol problem on time-p erio d [0 , T − ε ]. W e choos e the maximisatio n of the exp ected total yield of seeds ov er a lifespa n as the fitness measur e. Thus: Z T t 0 L ( s ) ˙ x 2 ( s ) ds = Z T t 0 L ( s )( v ( s ) − v 1 ( s )) g ( x 3 ( s )) ds → max . (4) A plant can maximise fitness b y c ho osing a n a ppropriate germination time, t 0 , and controls v and v 1 defined on [ t 0 , T ]. W e a ssume that the functions on the r ight-hand s ide of equations (1) are smo oth enough to guara nt ee the ex is- tence and uniq ueness o f solutions for (1). W e als o assume that the system (1) is forward complete, i.e. fo r all ini- tial conditions a nd all admissible c ontrols, the solution of (1) e xists for all time. F rom the biolo gical viewp oint this means that it is imp ossible to achiev e endless yields ov er a finite amo un t o f time, which in turn ensures that the solution to the pr oblem (1), (2), (4) exists. 2.2. Mo del pr e dictions W e p erform an ana lysis of (1), (4) with the help of Pon- tryagin’s Max im um Principle (PMP) [15], with the aim of determining in what order the p erio ds of a plant’s life follow ea c h other . The full analysis is presented in Ap- pendix 1, but for now we summarise the results der ived therein. First, we construct a Hamiltonian H , corr esp onding to the pro blem (1), (4), which is equal to t he scalar pr o duct of the rig h t-hand side o f (1) times the functional co efficients p 1 , L , p 3 . H = p 1 ( t ) ( v 1 ( t ) g ( x 3 ( t )) − µ ( t ) x 1 ( t )) + L ( t ) ( v ( t ) − v 1 ( t )) g ( x 3 ( t )) + p 3 ( t ) ( ζ ( t ) f ( x 1 ( t )) − v ( t ) g ( x 3 ( t )) − ω ( t ) x 3 ( t )) . The co efficients p 1 and p 3 are so -called ‘adjoint functions’ to the equa tions, a s they g ov ern the dynamics of x 1 and x 3 . Their dynamics is descr ibed by equations (11). The developmen t of a plant acco rding to the mo del (1) inv o lves three main p erio ds: (V) V egetative p erio d, during which ho lds p 1 ( t ) > max { L ( t ) , p 3 ( t ) } . (R) Repr o ductive p erio d, characterised by L ( t ) > max { p 1 ( t ) , p 3 ( t ) } . (S) Stor age p erio d, during which p 3 ( t ) > max { p 1 ( t ) , L ( t ) } . Thu s, the c hoice o f a co mpartment to which a plant should allo cate av ailable resourc es depends on the relatio n betw een functions p 1 , L , p 3 : all res ources should b e allo- cated to vegetative par ts if p 1 is the lar gest of the trio, to storage if p 3 is the la rgest and to repro duction if L is the largest. The perio ds V , R , S can be further sub divided into sub- per io ds which follow each other , as depicted in Figure 1: • D - Seed dorma ncy . • S. 2 - Pr eparing for unfav ourable climate co nditions. 3 • S. 1 - Life in unfav ourable c limate conditio ns. • V . 2 - V egetative p erio d that starts close to the end of the p erio d with unfav o urable conditions. At the beg inning of this p erio d, a plant starts allo cating to vegetativ e tissues as prepara tion for climate condi- tions which a re fa vourable for photosynthesis. • V . 1 Allo cation to vegetativ e tiss ues b efor e repro duc- tion. Impo rtant sp ecial cases o f the g eneral scheme are : 1. Annual plant with the po ssibility of multiple r epro- duction p erio ds (Figure 1 B). Multiple repro duction per io ds may appear if losses of vegetativ e mass due t o external factors mo delled b y the f unction µ ar e severe. This pa rticular case was analys ed in the ea rly work [18]. W e show a numerical example of this scenar io in Section 3.1. If µ ≡ 0, then mu ltiple repr o ductive per io ds for annual plants are not p ossible. 2. Mo no carpic plants. A sufficient (but not neces sary) condition for a pla n t to b e mono ca rpic (if there is no mortality) is the negligibility o f ω (in par ticular, if ω ≡ 0); in other words, the mass of stor age cannot decreas e due to external fa ctors. In this ca se, transitio ns R → S. 2 and V . 1 → V . 2 are not possible, as shown in Figure 1 C. W e show a numerical example of this scenario in Section 3.2. 3. Numerical examples In this section we present exa mples of numerical solu- tions for the mode l (1 ), which represents v a rious patterns of plant life histor ies. The examples a re not intended to mimic any specific pla n t sp ecies but to show that the mo del is adequate enough to include a broad range of strategies previously modelled separately . Additionally , we sho w that parameter changes may lead to qua litatively different results. W e star t with ann ual plan ts with single or m ultiple peri- o ds of repr o ductive allo cation, following which we pres ent the cases o f mono carpic and ev ergreen poly carpic plan ts as well a s a p olycarpic plant losing almost all of its v egetative parts but reta ining its storag e during unfav ourable sea- sons. Finally , we show that a nn ual o r mono carpic strate- gies can evolv e under cer tain conditions , even when lifes- pan is no t pr edefined. In all exa mples time is meas ured in months, with the be- ginning and the end of the simulation placed in the mid dle of a winter. The mass of all three co mpartments (vegeta- tive mass, s torage, r epro ductive mass) is meas ured in e n- ergy units, say MJ, to av oid using co efficients, in order to take into acco un t water conten t and differences in ener gy density betw een co mpartments. Seeds of the size 0.3 units contain 95 p er cent of s torage a nd 5 p er cent of vegeta- tive ma ss. The pho tosynthetic ra te is descr ibe d via the following saturatio n function: f ( x ) = ax bx + k , (5) where a, b, k > 0 ([2, p. 2 24]). Ano ther r easonable choice could b e an allometric function ax b , used in particular within Metabo lic Theory of Ecolog y (MTE), Dynamic En- ergy Budg ets (DEBs), etc. (see [19], [20]). W e mo del the depe ndence o f the photo synthetic rate on climate as follows: ζ ( t ) := 0 . 2 + 0 . 8    sin  π 12 t     . The maximal release r ate o f storag e tissues is linear, i.e. g ( x 3 ) = cx 3 for some c > 0 – while the actual release rate dep ends on the control v aria ble v ( t ). Stora ge and vegetativ e mass losses are defined separately for particular cases. If the contrary is not mentioned e xplicitly , we assume that there is no morta lit y ( L ≡ 1). Computations are made in Matlab, with the help of the optimal co nt rol solver GPOPS. 3.1. A nnual plants W e start with the s imple scena rio o f an annual plant with a single r epro duction p erio d at the end of the se ason. The r esults o f the s im ulation are depicted in the left-hand column of Figur e 2. A seed stays dormant for se veral weeks and then ger - minates using res ources from stor age. Germination takes place when environmental conditions a re still harsh, to pre- pare a pla n t for vegetative g rowth when these conditions improv e. Next, there is a phas e of pure vegetativ e growth, when a ll a ssimilated r esources a re allo ca ted to vegetative mass. After an instantaneous switch, a ll resources a re al- lo cated to r epro ductive ma ss a nd vegetativ e mass decays, reaching a very small v alue at the end o f life equal to 12 months. If v egetative mass losses are hea vier and cyclic, for example through r epea ted grazing, after the vegetative growth phase there are multiple switches b etw e en vege- tative a nd repro ductive allo ca tion, as shown in the rig h t- hand column of Figure 2. This is reminiscent of an early mo del [1 8]. In the fina l s tage a ll reso urces are allo cated to repro ductive ma ss. Figure 2 E a nd F show the dynamics of costate v aria bles (see Appendix 1 ) as well a s o f a sur viv ability function L (whic h is constant in these examples) fo r annual plants with one repro ductive p erio d and for annuals with multi- ple r epro ductive p erio ds. T a king into ac count that L re- places the costate v ar iable in the ca se of the repr o ductive compartment, we can say that resources s hould be dev oted to the compa rtment with the highest co state v ariable. In Figure 2 E we se e that if p 3 is not maximal o ver some time span, then it is only slightly smaller than the maximal costate v a riable on this in terv a l. T o ex plain this phenomenon, let us a ssume that at a given time t , a llo- cation to repro ductive tissues is o ptimal. If a plant do es 4 A: p er ennial plants D V S. 2 S. 1 V . 2 V . 1 R B: annual plants D V V . 1 R C: mono ca rpic pla n ts D V S. 2 S. 1 V . 2 V . 1 R Figure 1: Stag es of developmen t o f a p erennial (A), annual (B) and mono car pic (C) plant. D stays for seed dor mancy , V – vegetative gr owth after ger mination, S .2 – Prepa ring fo r unfav ourable c limate conditions , S .1 – life in unfav o urable climatic conditions, V .2 – vegetativ e p erio d, V .1 – allo cation to vegetative tissues b efor e r epro duction, R – r epro ductive allo cation. not behave optimally and do es not allo c ate to r epro duc- tive tissues at t , r esources are re tained in stor age and can b e a llo cated from stora ge to repro ductive tissues a bit later. Such a small delay decreas es fitness minimally , which means tha t the ‘usefulness’ of allo c ation to stor age is o nly slightly smaller than the ‘usefulness’ of allo cating to vegetative tissues. T o enhance the difference be t ween the cos tate v ar iables for storag e a nd repro duction (Fig- ure 2 F), we assumed in the annual plan t with multiple re- pro ductive p erio ds mo del that storag e losses increase over time. Life span was set to 12 months for the cases illustrated in Figur e 2. T o show why a nn ual strateg y could evolve, we s et the lifespan to several years but changed ω , de- scribing stor age losses: ω is now v ery low during almost the entire season, but it increa ses rapidly tow ard the end of the first fav ourable season. In such a case, the entire amount of storage is relo ca ted to repro ductive mass, while only remnants of vegetative mass survive thr ough to the next sea son and there is no vegetativ e g rowth in that sea- son (Figure 3 A). This is seemingly only virtually annual strategy , but we c an easily imagine that a real plan t w ould relo cate a ll mov able hydroca rb ons to r epro ductive mass befo re win ter and then die, b eca use remnan ts o f vegetativ e mass a re practically useless. Such relo catio n, co nsidered by [21], was no t a llow ed in the mo del. 3.2. Mono c arpic plants Our next scenario inv olves a mo no carpic pla nt with a five-y ear lifecycle. The r esults are depicted in Figure 4. Since a pla n t s tarts its development during winter, it do es not gro w b ecause the los s o f storag e is m uch low er th an t he loss of vegetative tiss ues. A t the b eginning of s pring we see the r egrowth of the plant from s torage, and then vege- tative growth due to photosy n thesis, which is followed by allo cation to stor age as prepara tion for winter. This pat- tern of developmen t c ontin ues until the last season, which ends with repro duction. An increase in parameter ω leads to the earlier regrowth of a plant from s torage. In the limit it leads to the ever- green pe rennial, as shown in the next subsection. Life spa n was set to 60 months for the case illus trated in Figur e 4. T o show why a mono car pic str ategy could evolv e, we set the lifespan to several years, a nd although small a t the b eginning, ω incr eases rapidly with a ge. Con- spicuous inflorescence of mono carpic plants may attract not only p ollinators but a lso enemies, which may ca use a rapid increase o f ω with the onset o f flow ering [22]. Now the first sea son is used for vegetative g rowth a nd then for building storage (Figure 3 B). In the second year, reg rowth from stor age a nd then growth fr om assimila ted r esources take place, followed by complete allo ca tion to repro duc- tive mass . Although remna n ts of vegetative mass survive through to the ne xt seas on, we can call the plant a mo no- carpic biennia l b ecaus e there is no vegetativ e g rowth in the third year. 3.3. Per ennial p olyc arpic plants Figure 5 presents n umerical exa mples of p er ennial p oly- carpic pla n ts. In the case shown in the uppe r row, plant mortality is neglected, but s torage losses are mo dera te. After a few years of pure vegetativ e growth, with the p eak size of a pla nt incr easing, there are years of mixed veg- etative and r epro ductive growth. Peak size is constant ov er several years a nd it decrea ses close to the end of life, 5 A: Annual with o ne re pro duction p erio d: s tates 0 1 2 3 4 5 6 7 8 9 10 11 12 −1 0 1 2 3 4 5 6 7 t ( x 1 ( t ) , x 2 ( t ) , x 3 ( t )) x 1 ( t ) x 2 ( t ) x 3 ( t ) B: Ann ual with multiple repro duction p er io ds: s tates 0 1 2 3 4 5 6 7 8 9 10 11 12 −2 0 2 4 6 8 10 12 14 16 t ( x 1 ( t ) , x 2 ( t ) , x 3 ( t )) x 1 ( t ) x 2 ( t ) x 3 ( t ) C: Ann ual with one repro duction p erio d: controls 0 1 2 3 4 5 6 7 8 9 10 11 12 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 t v 1 ( t ) , v 2 ( t ) v 1 ( t ) v 2 ( t ) D: Annual with multiple r epro duction per io ds: controls 0 1 2 3 4 5 6 7 8 9 10 11 12 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 t v 1 ( t ) , v 2 ( t ) v 1 ( t ) v 2 ( t ) E: Annual with o ne repr o duction p erio d: costates 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 t p 1 ( t ) , L ( t ) , p 3 ( t ) p 1 ( t ) L ( t ) p 3 ( t ) F: Ann ual with m ultiple repro duction p erio ds: co states 0 1 2 3 4 5 6 7 8 9 10 11 12 0 0.5 1 1.5 2 2.5 3 3.5 t p 1 ( t ) , L ( t ) , p 3 ( t ) p 1 ( t ) L ( t ) p 3 ( t ) Figure 2: Annual pla n t with sing le (left co lumn) or multiple (rig h t column) repr o duction p erio ds. A and B are the states, C and D are the cor resp onding controls and E and F ar e the costate v ariables. The maximu m photosy n thesis rate f ( x ) = 1 . 5 x 1+0 . 3 x and the stor age relea se r ate g ( x ) = 5 x , where x is vegetativ e mass. The rate of destructio n of vegetativ e tissues µ ( t ) = 0 . 8   cos  π 12 t    for the left-ha nd co lumn and µ ( t ) = 1 . 8 | cos ( π t ) | fo r the right-hand column. The rate of stora ge lo sses ω ≡ 0 . 05 for the left co lumn; although it was p ossible to obtain multiple switches fo r the same ω , this function has b een c hanged in the right-hand co lumn to ω ( t ) = t 1+0 . 5 t to better visualise the difference in costates betw een stora ge and repro ductive output. 6 A: No n-forced annual: states 0 6 12 18 24 30 36 42 48 54 60 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 t ( x 1 ( t ) , x 2 ( t ) , x 3 ( t )) x 1 ( t ) x 2 ( t ) x 3 ( t ) B: Non-fo rced biennia l: sta tes 0 6 12 18 24 30 36 42 48 54 60 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 t ( x 1 ( t ) , x 2 ( t ) , x 3 ( t )) x 1 ( t ) x 2 ( t ) x 3 ( t ) Figure 3 : Optimality of a nn ual (A) and biennial (B) str ategies, non-for ced by limited lifespan T . Storage lo sses are negligible at the b eginning o f life and increase v ery rapidly thereafter. The maximum photosyn thesis rate f ( x ) = 0 . 5 x 1+0 . 01 x and the stor age relea se rate g ( x ) = 2 . 5 x , where x is vegetative mass. The rate of des truction of the vegetative tis sues µ ( t ) = 0 . 8   cos  π 12 t    for the annual and µ ( t ) = 0 . 3   cos  π 12 t    for the biennial. The rate of stora ge lo sses equals ω ( t ) = 0 . 00 0002 t 6 for the a nn ual and ω ( t ) = 0 . 00000 05 t 5 for the biennia l. A: Mo no carp: states 0 6 12 18 24 30 36 42 48 54 60 −5 0 5 10 15 20 25 t ( x 1 ( t ) , x 2 ( t ) , x 3 ( t )) x 1 ( t ) x 2 ( t ) x 3 ( t ) B: Mono carp: controls 0 6 12 18 24 30 36 42 48 54 60 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 t v 1 ( t ) , v 2 ( t ) v 1 ( t ) v 2 ( t ) Figure 4: Mono ca rp plant with a lifespa n set to 60 months. State v ariables a nd controls ar e depicted on A a nd B, resp ectively . The maximum photosynthesis rate is f ( x ) = 0 . 5 x 1+0 . 01 x and the stor age release r ate equals g ( x ) = 2 . 5 x , wher e x is vegetativ e mass. The rate of destruction of the vegetativ e tissues is µ ( t ) = 0 . 4   cos  π 12 t    and the rate of storag e losses ω ≡ 0 . 1. 7 which is arbitrar ily set a t 96 months. Without such lim- itation, yearly cycles would b e rep eated infinitely . Each cycle starts with regrowth fr om sto rage during the end phase of an unfav ourable seaso n, then the g rowth o f veg- etative par ts fr om the current photos ynth esis, follow ed by an instantaneous a nd complete switch to r epro ductive al- lo cation (if it app ear s), followed by the instantaneous a nd complete switch to building s torage. In par allel, vegeta- tive mass decrea ses during b o th r epro ductive allo cation and building storag e, and this decline contin ues dur ing a n unfav o urable season. Only a small amount of vegetative mass p ersists ov er an unfav ourable sea son, but s tored re- sources allow for quick re growth in the next season. Note that the size of stor age which is o ptimal for flow ering is reached in the wint er b efore the fir st repr o duction phas e. The middle row in Figur e 5 r epresents an evergreen plant. Here, sto rage loss es are so high and losses of veg- etative mass ar e low eno ugh so that a plant can survive during an unfav ourable seaso n in the form of vegetative parts and storag e is no t b eing built. B ecause the v ariable x 3 represents not only s torage but also free suga rs still not allo cated to a nother co mpartment, x 3 is no t ex actly equa l to zer o. Note that ess en tially the pattern of growth o f an e vergreen p erennial plant res em bles the life histo ry of an annual with mult iple repro ductive p erio ds, ex cept for longer multi-y ear life. The lowest row in Figure 5 repr esents a p erennia l plant that does not lose stora ge but is sub ject to a constant mor- tality rate (i.e. exp onentially d ecreas ing surviv al probabil- it y L ( t )). T o show that mortality has a qua litative effect similar to s torage los ses, we cho ose the case with ω = 0. A plant sub ject to b oth mortality and s torage losses will achiev e sma ller size, but the gener al pattern will b e the same. Repro ductive output is very high in the illustrated case, bec ause it r epresents pla n ts tha t s urvive to the final time T . F or fitness ca lculation, x 2 is weigh ted by L . 4. Optimis ation o f see d mass In the previous sec tions we defined the fitness o f a pla n t as a n exp ectatio n of the mass of repr o ductiv e tissues pro- duced by the plant during its lifetime. T o maximise fi tness, a plan t co nt rols the allo cation of photosynthate and g er- mination time. Howev er, it is well-kno wn tha t for plants that pr opagate exc lusively through seeds, fitness dep ends crucially on the quantit y and size o f the s eeds a plant pro- duces. Current mo dels of optimal allo cation do not pro- vide this infor mation, and so the ma ss of a seed is treated as an exter nal para meter. E ssentially , though, choosing the mass of seed is an additiona l control which a plant can use in o rder to allo cate the photosy n thate efficiently . Therefore in this s ection w e extend the mo del fr om the previous one by giving the plant additional co nt rol ov er the mass o f s eed. Let y 0 = ( y 0 1 , y 0 2 , y 0 3 ) b e the total ma ss of seeds (the vector consis ting of the masse s o f three compo nents of a plant) that has to b e divided b etw een a seeds, while a ∈ [1 , ∞ ) and a can b e either natural or r eal num ber . W e a ssume that the mass of each seed is s = y 0 a . The equations determining the dynamics of a plant a re as fo llows: ˙ x 1 = v 1 ( t ) g ( x 3 ) − µ ( t ) x 1 , ˙ x 2 = ( v ( t ) − v 1 ( t )) g ( x 3 ) , ˙ x 3 = ζ ( t ) f ( x 1 ) − v ( t ) g ( x 3 ) − ω ( t ) x 3 , x (0) = 1 a y 0 . (6) Here x (0) = ( x 1 (0) , x 2 (0) , x 3 (0)). W e lo ok for the v alues o f co n trol v ariables that max- imise the total mass of repro ductive tissues pro duced by all dir ect descendants: max 0 ≤ v ( t ) ≤ 1 , 0 ≤ v 1 ( t ) ≤ v ( t ) , a ∈ [1 , ∞ ) Q a = aξ Z T t 0 L ( s ) ˙ x 2 ( s ) ds, (7) where a cons tant ξ is the fr action of germina ting seeds. W e a ssume that ξ do es no t dep end on the size of a se ed. In contrast to the problem (1), (4) with the fix ed mass of the seed, the problem (6), (7) may have no s olution, in the sense that every finite amount o f s eeds w ill not b e op- timal, which means (with a s light abuse of mathematical rigoro usness) that the size of the seeds should be infinitely small ( a → ∞ ), i.e. no admissible controls v , v 1 and pa- rameter a g enerate the optimal v alue of Q a . Recall that a function f is ca lled concav e on the set M , if ∀ z 1 , z 2 ∈ M , ∀ α ∈ [0 , 1] a n inequality f ( αz 1 + (1 − α ) z 2 ) ≥ αf ( z 1 ) + (1 − α ) f ( z 2 ) (8) holds. If the inequality ( 8) holds with ≤ instead of ≥ , then the function f is called conv ex on set M . F r om biological vie wpo in t imp or tant is the c ase when f and g are conca ve functions, i.e. the rate of photos ynt hesis and the maximal speed of c hemical reactions in a plant a re saturated as a result of the gr owth of the mass of a plant (due to self-shading o f leaves, nutrien t depletion in the soil, etc.). F or such functions, together with co ndition f (0) = g (0) = 0, a ∈ [1 , ∞ ), one can prov e (see Appendix 2) that Q a increases when a increa ses. It follows from this result that when f and g are concave, the b est strategy for a plant is to pro duce as many seeds as p ossible, whic h means that the seeds should b e a s small as po ssible. A similar arg umen t shows that for convex functions f , g , the o ptimal mass of the seed has to b e as large a s p ossible (without additional restrictions on the quantit y of seeds a = 1). The b oundar y case o ccurs when b oth f and g a re linear functions, which are concav e and con vex at the same time. In this case it follows from (20) that the yield o f a plant do es no t dep end on the mass of the seeds. Another obs erv ation is that if f and g ar e concav e and contin uo usly different iable in the neighbourho o d of 0 , then the solution of the linea rised model (6), (7) provides the ‘theoretical’ upp er b ound for the fitness of the plant under consideratio n. 8 A: Perennial pla nt : states 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 −1 0 1 2 3 4 5 6 7 8 t ( x 1 ( t ) , x 2 ( t ) , x 3 ( t )) x 1 ( t ) x 2 ( t ) x 3 ( t ) B: Polycarpic plant: co nt rols 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 t v 1 ( t ) , v 2 ( t ) v 1 ( t ) v 2 ( t ) C: Evergreen po lycarp: states 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 −10 0 10 20 30 40 50 t ( x 1 ( t ) , x 2 ( t ) , x 3 ( t )) x 1 ( t ) x 2 ( t ) x 3 ( t ) D: E vergreen p oly carp: co n trols 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 t v 1 ( t ) , v 2 ( t ) v 1 ( t ) v 2 ( t ) E: Polycarp with mortality: sta tes 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 −5 0 5 10 15 20 25 t ( x 1 ( t ) , x 2 ( t ) , x 3 ( t )) x 1 ( t ) x 2 ( t ) x 3 ( t ) F: Polycarp with mor tality: controls 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 t v 1 ( t ) , v 2 ( t ) v 1 ( t ) v 2 ( t ) Figure 5: Perennial plan t, with mainly storage per sisting o ver an unf av ourable season (A, B, E, F), and an e vergreen with per manent v egetative parts and no s torage (C a nd D). In A-D morta lit y is neg lected, but storag e losses are mo der ate for A and B ( ω ≡ 0 . 1 5) or hig h for C and D ( ω ≡ 1 ). In E and F storage losses are neg lected, but a constant mo rtality ra te of 0.03 is as sumed (probability o f surviv a l to age t , L ( t ) = e − 0 . 03 t ); note that the results fo r A-B and E-F a re qualitatively similar. A, C a nd E are the states a nd B, D and F are the co rresp onding controls. The maximum photo synthesis rate f ( x ) = 0 . 5 x 1+0 . 1 x and the maximum storage r elease ra te g ( x ) = 2 . 5 x , where x is vegetativ e mass. The rate of de struction of vegetativ e tissues µ ( t ) = 0 . 4   cos  π 12 t    for A, B, E and F and µ ( t ) = 0 . 1   cos  π 12 t    for C and D. 9 5. Discussion Cohen [1 ] pro duced the first explicit mo del for optimis- ing the allo c ation o f limited resour ces to growth o r repro - duction, a nd this model was applies to annual plants, with the photo synthesis ra te dep endent linearly on vegetativ e mass. Denholm [2 3] confirmed Cohen’s result through the PMP metho d. The mo del was gener alised to a non- linear photosynthesis rate by Vincen t and Pulliam [2] a nd Zi´ o lko and K oz lowski [24]. Both of these pap ers used the PMP metho d to analyse the problem, and in the last one mortal- it y was int ro duced to the mo del. King a nd Ro ughgarden [18] also considered vegetative mass loss es and sho w ed that m ultiple switches fro m v egetative to r epro ductive allo ca- tion may b e optimal if thes e losses are heavy – the r esult that we were a ble to repro duce in our gener al mo del. The first paper s on o ptimal a llo cation in pe rennial plants ap- pea red la ter in [2 5], [2 6] and [13]. Our mo del combines several sp ecific mo dels within the same fra mew ork, which is a g reat adv a nt age. This goal was achiev ed through a crucial m o dification in comparison to the pr evious mo dels: photosynthates a re not allo cated directly to vegetative and re pro ductive tissues, but firstly to stor age, fro m which they c ould b e relo c ated to other compartments, if o ptimal at a given time. Such a redefini- tion of a s torage co mpartment, also including suga rs just pro duced, is biologica lly rea sonable a s well as fruitful for mathematical mo delling. Analytica l a nalysis o f the mo del leads to the de velopmen t of g eneral schemes o f life his- tory phases, illus trated in Figure 1 . Such schemes will b e very us eful in building more adv anced mo dels, as well as in planning field studies. The analysis o f the mode l con- firmed that o ptimal switches are instantaneous, in other words resour ces should b e a llo cated to only one compar t- men t a t a ny given time t . Resources should go to the vegetativ e compar tmen t if p 1 is la rger than L and p 3 , to storage if p 3 is larger than L and p 1 and to r epro duction if L is the la rgest of the three. Simultaneous alloc ation would be optimal only if tw o (o r more) of functions p 1 , L , p 3 were equal, which never app ear s in the prese n ted mo del. How- ever, we showed in numerical examples tha t the difference betw een costate v ariable for storag e p 3 and a maximum of L a nd p 1 is fairly s mall, which indicates that the pr ice for sub optimal strategies ca n b e very low. Thus, we s hould not ex pec t strictly bang-bang switches in na ture. So me constraints can also for ce the o ptimalit y o f non bang- bang solutions, as discussed la ter in this pap er . Apart from the m ost imp or tant no velt y , i.e treating stor- age a s the primary sink for photosynthates, we added s ev- eral o ther novelt ies to the mo del. Seasonality was mo d- elled by changing the photosynthesis r ate, vegetativ e tis- sue lo sses and storag e lo sses ov er time. Although we used simple p erio dic functions in o ur n umerical examples, func- tions extracted fro m r eal data could be applied as well. Such functions could take into acco un t a common in some geogr aphical regio ns mid-summer depres sion, i.e. decreas- ing the rate of photosynthesis in summer months, usually caused b y water limitation (e.g. [27]), as well a s ear ly de- creases in lig ht pe netration down to the forest flo or , which forces some p erennia ls to blo om and to s et se eds ear ly in spring. In o ther pap ers o n optimal allo cation, seasonal changes have no t b een g radual: after a fav ourable season, the onset of win ter is rather abrupt, and then the next fav o urable seas on app ears instantaneously . Such an ap- proach precludes discov ering an interesting phenomenon: under some assumed stor age lo sses and vegetativ e pa rts, germination or spring regr owth ma y still a ppea r in an un- fav o urable s eason, which allows for us ing the pho tosyn- thetic p otential of a fav ourable part of the sea son in full. If losses of vegetative parts in winter are lo w, an evergree n strategy without pro ducing s torage is optimal, as shown in Figur e 5 C-D. Such p erennials a re common, but ea r- lier mo dels were no t able to show this strateg y b eca use of the non-rea listic treatment of seasona lit y . Note that the transition from the strategy of p ersisting ov er winter in the form o f almost an e xclusive stora ge or gan to an ev- ergreen strateg y with virtua lly no storag e is gra dual – if we decreas e the losse s of vegetativ e tissue with r espe ct to storage losse s, regr owth fro m stora ge b ecomes gradually earlier. Sto rage los ses were not consider ed ear lier in allo- cation mo dels, except in the co n text of storage as a ba ckup for an unpredicta ble a nd ba sically a -seaso nal en vironment [28]. Although we do not show s uc h a result, annuals starting to gr ow in autumn and c ontin uing the pr o cess throug h to spring, as in the cas e of winter cerea ls, would be o ptimal under so me sp ecific los s functions for vegetative par ts. The interpretation of so me of the terms used in this pap er co uld b e more genera l. As sta ted pre viously , ‘stor- age’ ( x 3 in the mo del) contains real s torage in the for m o f starch or fat, as well a s free ca rb ohydrates just pro duced. Similarly , los ses of v egetative par ts may include no t only grazing o r the decaying of leav es, but a lso a dec rease in photosynthetic po ten tial typical of age ing leaves. Stora ge losses may mean a usage of a par t of s torage for metab olic pro cesses, but also partial consumption of s torage o rgans by a nimals. Our mo del, as each mo del, has its limita tions. Assuming that f dep ends o nly on vegetative mass ( x 1 in the mo del), we neg lect the po ssibility of photos ynt hetic a ctivity in re- pro ductive tissues. Suc h a p ossibility was considered by King and Roughgarde n [18], a lbe it only for annual plants. Since we s uppos e that g is a function of x 3 , the rate o f allo cation from stora ge to vegetative and/or r epro ductive tissues is limited only by the mass of stora ge. This lea ds to the rapid reg rowth of a plant from stora ge, b ecause the allo cation rate can b e very high, even if the mass of exist- ing vegetativ e tissues is extre mely s mall. T o exclude this effect one could consider mo re gener al functions g , depend- ing also o n x 1 , x 2 . Another po ssibility , inv estigated in [4], [5], is to intro duce a dditional constr aints to the allo cation rate from one type of tissue to ano ther. How ever, this is- sue is o utside of the scop e of this pap er. W e also do no t allow for the r elo cation of reso urces from vegetative mass 10 to repro duction or storage , which co uld b e optimal when the rate o f pho tosynthesis dr ops to a very low level [21]. An imp ortant and spec ial case for p er ennial plants ar e mono carps. A t present , no unified theor y expla ins the origins of mono carpy [42, 35], but in this pap er we pr o- vide sev eral p ossibilities for the emergence o f a monoca rpic strategy . The fir st sufficient condition for the genes is of a mono- carpy , s hown in App endix 1, is the negligibility of mo r- tality and s torage los ses. In this case the s trategy is not forced by the choice of the length of life T , and q ualita- tively it do es not dep end on the choice of functions f and g . In this case for a plant there is no sense in repr o ducing earlier b eca use it ca n save e nergy in the stor age and, s ince storage c annot b e lost, use it later fo r pro ducing eith er veg- etative or repro ductive tissues. One may argue that the mortality of a parental plant usua lly cannot b e neglected for p erennia l sp ecies, but if we a lso add losses o f repro- ductive tissues, then the negligibility of storage loss es ma y still lead to the development o f a mono ca rpic lifecycle. Mono carps (or annuals) may also a rise from our mo del when storage lo sses ar e mo derate, exe mplified in Figures 2 and 4. Ho wev er, these strateg ies are in a sense forced by limited life s pan, a s the pla n ts hav e no ch oice but to repro duce at the end of their life. Such a scenario is b e- liev able if certain constraints are no t a llowing the pla n t to live lo nger; for exa mple, if a pla n t sp ecies has no mo rpho- logical or ph ysiolog ical mechanisms allowing it to survive during the winter, it mu st be annual. How ever, we can find many plant gener a in w hic h annual, mo no carpic and po lycarpic sp ecies co exist. Even more so, some sp ecies r ep- resent different strategies in different r egions, whic h means that constraining to one str ategy is infreq uen t. A third p ossible scenar io which leads to mono c arpy is a rapid incr ease in stor age lo sses over time. As w e show in ex amples illustrated in Figure 3 , even without a limited lifespan a n annual o r mo no carpic strateg y ca n evolv e if storage losses or mortality rapidly increase. W e ca ll these cases no n-forced a nn uals a nd mo no carps. In annuals, pr o- ducing resista nce to frozen storage may have high over- head cos ts, and necess ary resources may b e more efficiently used for additional seed pro duction. Mo no carps must b e equipp ed with mechanisms allowing for winter surviv al, so we ther efore hav e to seek a nother explanation. De Jong and Klinkhamer [22] sugg est t hat mas s bloo ming of mono- carps attra cts enemies. F or example, flowering Sene cio ja- c ob ae a plants ha ve t wic e a s many T yria jac ob ae ae butterfly egg ba tc hes than non- flow e ring plan ts. Losses o f vegetativ e and stora ge or gans would b e so high that pro ducing addi- tional seeds would increas e fitness f ar more than storing re- sources for further life. Similarly , Cygno glossum officinale setting s eed pla n ts are a lmost always attac ked b y the r o ot weevil Ceutorhynchus cru ciger , whereas non-r epro ducing plants ar e a lmost never a ttack e d. T o explain the devel- opment o f mono carpy fo r such spe cies, one may consider storage losse s, dep ending on a mas s of repro ductive tissues, as do ne in [4 2]. It is p ossible to include such b ehaviour in our mo del by taking ω = ω ( t, x 2 ). W e hop e that an a nal- ysis o f this gener alisation of our mo del will provide new insights in to the g enesis o f mono carpy in p er ennial pla nt s. W e note that the optimal size of a seed depends crucially on the f orm o f the functions f a nd g . F or concav e functions that are often us ed to take into account self-sha ding, the bo undedness o f resourc es, etc. (see e.g. [13]), w e hav e prov ed accor ding to our mo del that seeds hav e to b e as small a s p oss ible. F o r plan ts living in ope n environments and for spe cies o ccupying ea rly phases in successio n (colonising s pecie s), the assumption of concavit y is not an ov ersimplification. The behaviour that o ur model predicts, namely that the optimal strateg y is to pro duce a v ast amount o f small seeds, is typical fo r these sp ecies [29]. How ever, in clos ed and sha dy environmen ts, under mineral s hortage condi- tions, or if there is strong co mpetition from established vegetation, the rate of photosyn thesis per unit of mass can increase with the increasing mass of the plant, i.e. function f is c onv e x on some [0 , p ], p > 0 and the seeds cannot be to o small. These predictio ns are, in g eneral, in acco rdance with e xpe riments [3 0] a nd [3 1], but see [32]. Note that a similar r esult for offspr ing size in animals was obtained by T a ylor and Williams [33] and Koz lowski [34]: in a -seasona l environments, offspring should b e as small as p ossible if the dep endence of pro duction rate/mor tality r ate is concav e, and they should hav e some optimal s ize if dep endence is conv ex. Because in their mo dels the nece ssary condition for the e xistence of opti- mal adult size is the c oncavit y of the ratio, optimal size for b oth adults and offspring can app ear only if the ratio has a n inflection p oint(s). Another imp ortant ques tion is a ssessing how rea listic are fitness meas ures (4) a nd (7) . Such measures of fitnes s, known also as ‘lifetime offspring pro duction’, a re rea son- able o nly in stable populatio ns. In p opulations chang- ing their size, the timing of repro ductio n is also imp or- tant. Descendants which have b een pro duced ea rlier c an in turn pro duce their own offspring earlier , and thus they are more v aluable than de scendants pro duce d at a later stage. This indica tes that the r ight measure of fitness o f the plant is not the pro duction of offspring over a lifetime, but lifetime offspring pro duction b y this plant and its de- scendants. This leads to a co nsiderably mo re complicated (and more challenging) optimal control mo del, which to our knowledge has no t b een a ddressed in the literatur e for contin uo us-time mo dels of plants. Much mo re frequent is a simpler version of fitness, defined as a num ber of descen- dants dis counted to the pr esent [3 6]: Z T t 0 W ( s )( v ( s ) − v 1 ( s )) g ( x 3 ( s )) ds → max , where W ( s ) = L ( s ) e − r s , for all s ≥ 0, r is the p opula tion growth rate [37], [38], [3 9]. If we assume that r ≥ 0, then W is a decr easing function o f time with W (0) = L (0), a nd an analysis of life-histories – as p erformed in this pap er – 11 is still v alid for fitness measures (5 ) if we c hange L to W . Despite some limitations, our mo del, which unifies pre- vious sp ecific a llo cation mo dels, pro vides deep insights in to a broad range of p lant strategies. Res trictions on the func- tions f a nd g as well as on o ther functions descr ibing sea- sonality in (1) are v ery mild and allo w f or finding the right parametrisa tion of the mo del for a particular sp ecies or group o f sp ecies. This makes the mo del helpful for plan- ning exp eriments and field meas urements. 6. Conclusions and outlo ok In the present pap er we hav e developed a mo del that describ es the optimal allo catio n stra tegies of a p erennial (as well a s a n a nn ual) plant dur ing all the stag es of its life. T he mo del was analys ed with the help of Pon trya- gin’s Maximum Principle, and as a consequence we hav e derived a description of the life of a p erennial pla n t (Fig- ure 1 ) from dormancy un til its dea th. This mo del enco m- passes the mo dels of annual as well a s mono car pic plants. The sufficient condition for mono ca rpicity when there is no mortality has a lso b een pre sented. In Section 3 we have shown by means of numerical sim- ulations that the patterns of developmen t of annual plan ts with one o r mult iple repr o duction p erio ds, mono carpic, ev- ergreen and poly carpic p erennial plan ts, c an b e obtained through differen t choices o f mo del parameters. D ue to mortality and/or by incr easing the destruction rate of sto r- age, one can obtain ‘non-forced’ annuals as well as mo no- carps. In Section 4 we a nalysed the trade-off b etw een size and the num b er o f seeds for optimal allo cation problems. W e hav e provided s ufficien t conditions to ensure that an op- timal strategy will pro duce a s many (or as few) seeds as po ssible. The applicability of our results has be en discussed in Section 5. Although our mo del enco mpasses a wide v a riety of dif- ferent patter ns of pla n t development , it is ina pplicable to certain types o f plants. In particular , it see ms that it is hardly p oss ible to e xplain the life histor ies of plants whic h at the b eginning o f the seas on pro duce flow ers and only then s tart to develop vegetative tissues . It seems that comp etition for p ollinators should b e introduced to the mo del in order to ex plain such a strategy . Plants with vegetative reproductio n are entirely b eyond the sco pe of this mo del. In s pite of the commonness o f such plants in nature, o nly a few pa per s are devoted to studying their life strategies [40], [41]. The developmen t of a unifying mo del for plants with vegetative and s exual repro duction is also a challenging direction fo r r esearch. Another interesting topic inv olves g eneralising the r e- sults in Sectio n 4 to the case when the surv iv ability of a seed dep ends o n its size, which will b e more realistic from the biologica l po int of view. It seems that metho ds dif- ferent fr om thos e used in our pap er will b e required to address this problem according ly . W e hav e assumed in the pap er that mortality is age-dep endent, but size -indepe nden t. Int ro ducing siz e- depe nden t mortality into the mo del (1) ma kes a nalysis with the a id of PMP consider ably more complex. Due to a n um b er of r esults ac hiev ed with the help o f this mo del, a s well as due to a v ariety of p ossible dir ections for future in vestigations, we b elieve that this pap er is a go od starting p oint fo r a fr uitful res earch pr ogram in this area . Ac knowledgment s W e thank Filip Kapustk a and V olo dymyr Nemer tsalov for fr uitful discussions a nd constructive sug gestions. The computations hav e been ma de in Matlab, with the help o f the optimal control solver GPOPS. J K was funded by J agiellonian University (DS/WBiNoZ/INoS 757/1 3). App endix 1: Mo del analysi s F o r analysis of (1) we exploit Pon tryagin’s Maximum Principle (see, e.g. [15]). Note that we ca n dr op the eq ua- tion for x 2 , since the other equatio ns of (1) as w ell as the cost functional (4 ) after substitution o f ˙ x 2 do not dep end on x 2 . The Hamiltonian o f (1), (4) is de fined by: H = p 1 ( t ) ( v 1 ( t ) g ( x 3 ( t )) − µ ( t ) x 1 ( t )) + λ 0 L ( t ) ( v ( t ) − v 1 ( t )) g ( x 3 ( t )) + p 3 ( t ) ( ζ ( t ) f ( x 1 ( t )) − v ( t ) g ( x 3 ( t )) − ω ( t ) x 3 ( t )) . (9) Here λ 0 ≥ 0 and p 1 , p 3 are so-called adjoint functions. The eq uations determining their dynamics will b e given later. T o simplify the nota tion, we will frequently wr ite in equations simply p 1 , x 2 , etc. instead of p 1 ( t ), x 2 ( t ), if there a rises no a m biguity . W e r ewrite e xpression (9 ) in a more suitable form H = p 3 ζ ( t ) f ( x 1 ) − p 1 µ ( t ) x 1 + g ( x 3 ) ( v 1 ( t )( p 1 − λ 0 L ( t )) + v ( t )( λ 0 L ( t ) − p 3 )) − p 3 ω ( t ) x 3 . (10) Equations fo r the adjoint function p are as follows ˙ p 1 = p 1 µ ( t ) − p 3 ζ ( t ) ∂ f ∂ x 1 ( x 1 ) , ˙ p 3 = − ∂ g ∂ x 3 ( x 3 )( v 1 ( p 1 − λ 0 L ( t ))+ v ( λ 0 L ( t ) − p 3 ))+ p 3 ω ( t ) . (11) The corres po nding b ounda ry c onditions a re p 1 ( T ) = p 3 ( T ) = 0 . (12) If λ 0 = 0, then fro m (12) and (11) we obtain that p i ≡ 0 on [ t 0 , T ], fro m whic h it follows that all the controls ar e po ssible. Let λ 0 > 0. W e can take in this cas e λ 0 = 1. T o obtain the v a lues of v , v 1 , we so lve the pro blem H → max , 0 ≤ v ≤ 1 , 0 ≤ v 1 ≤ v . It is not har d to chec k that its solution is given b y 12 1. If L ( t ) − p 3 ( t ) > 0, then v ( t ) = 1, a nd v 1 ( t ) =    v ( t ) if p 1 ( t ) − L ( t ) > 0 , 0 if p 1 ( t ) − L ( t ) < 0 , ∈ [0 , v ] if p 1 ( t ) − L ( t ) = 0 . (13) 2. If L ( t ) − p 3 ( t ) = 0, then p 1 ( t ) − L ( t ) > 0 ⇒ v ( t ) = 1 , v 1 ( t ) = v ( t ) p 1 ( t ) − L ( t ) = 0 ⇒ v ( t ) , v 1 ( t ) - EAC p 1 ( t ) − L ( t ) < 0 ⇒ v ( t ) - EAC , v 1 ( t ) = 0 (14) 3. If L ( t ) − p 3 ( t ) < 0, then • if p 1 ( t ) − L ( t ) ≤ 0 then v ( t ) = v 1 ( t ) = 0. • if p 1 ( t ) − L ( t ) > 0 then p 1 ( t ) − p 3 ( t ) < 0 ⇒ v ( t ) = v 1 ( t ) = 0 p 1 ( t ) − p 3 ( t ) = 0 ⇒ v ( t ) - EAC , v 1 ( t )= v ( t ) p 1 ( t ) − p 3 ( t ) > 0 ⇒ v ( t ) = v 1 ( t ) = 1 (15) Here the abbrev iation EA C stands for ”every admissible control”. W e intro duce three main p erio ds characterize d by dif- ferent v alues of controls: (V) V egetative p erio d: p 1 ( t ) > max { L ( t ) , p 3 ( t ) } . In this case v ( t ) = v 1 ( t ) = 1, that is the vegetative pa rts are being constructed with the ma ximal r ate. Equations (11) in the vegetative p er io d take for m ˙ p 1 = p 1 ( t ) µ ( t ) − p 3 ( t ) ζ ( t ) ∂ f ∂ x 1 ( x 1 ( t )) , ˙ p 3 = − ∂ g ∂ x 3 ( x 3 ( t ))( p 1 ( t ) − p 3 ( t )) + ω ( t ) p 3 . (16) (R) Repr o ductive p erio d: L ( t ) > max { p 1 ( t ) , p 3 ( t ) } . In this case v ( t ) = 1 , v 1 ( t ) = 0 and repro ductive tissues are be ing constructed with the ma ximal r ate. Equations (11) within this p er io d ta ke the for m ˙ p 1 = p 1 ( t ) µ ( t ) − p 3 ( t ) ζ ( t ) ∂ f ∂ x 1 ( x 1 ( t )) , ˙ p 3 = − ∂ g ∂ x 3 ( x 3 ( t ))( L ( t ) − p 3 ( t )) + ω ( t ) p 3 ( t ) . (17) (S) Stor age perio d: p 3 ( t ) > max { p 1 ( t ) , L ( t ) } . In this case v ( t ) = v 1 ( t ) = 0 and all a llo cated energy go es to storage . The corres po nding equations (11) take the fo rm ˙ p 1 = p 1 ( t ) µ ( t ) − p 3 ( t ) ζ ( t ) ∂ f ∂ x 1 ( x 1 ( t )) , ˙ p 3 = ω ( t ) p 3 . (18) W e are g oing to ana lyze these p er io ds mor e deeply a nd find out in what o rder these p erio ds can arise in a life o f a plant. T o this end we investigate equations (11) from the end of the life o f a plant. Controls v and v 1 maximize the v alue of ( v 1 ( p 1 − L ( t )) + v ( L ( t ) − p 3 )), therefore for optimal v , v 1 it ho lds that ( v 1 ( p 1 − L ( t )) + v ( L ( t ) − p 3 )) ≥ 0 . (19) Note that in case, when ω ( t ) ≡ 0 (that is, if sto rage parts ca nnot b e destructed due to external factor s) this inequality and mono tonicity o f g imply that p 3 is an no n- increasing function o n [ t 0 , T ]. Let us analyze the behavior of Lagrang e multipliers p i and v alues of co nt rols a t the neig hborho o d of the time T . If the la st p er io d would b e vegetative, then the equa- tions, gov erning the dynamics of p 1 , p 3 would be (16). Due to w ellp osedness o f (16), and since the conditions (12) hold, we obtain, that p 1 ( t ) ≡ 0 and p 3 ( t ) ≡ 0 in the neig h- bo rho o d of time T . Since L ( t ) > 0 for all t < T , w e come to a contradiction with an assumption tha t the la st p e- rio d is vegetative. Analogously one can show that also the storage p erio d cannot b e the last p erio d of a plan t’s life. This proves, that the last p erio d of a plant develop ment is a r epr o ductive p erio d . F r om equations (19) and (11) using mono tonicity of g and inequality ω ≥ 0 we hav e that if for some τ ∈ [ t 0 , T ] p 3 ( τ ) < 0, then p 3 ( t ) < 0 for all t ∈ [ τ , T ], which contra- dicts to (1 2). Thus, p 3 ≥ 0 on [ t 0 , T ]. Analogously one can pr ov e that p 1 ≥ 0 o n [ t 0 , T ]. Now let us find out, wha t p erio d can precede to the re- pro ductive p erio d. Accor ding to equations (18 ) and due to ω ≥ 0 we see, that p 3 cannot decr ease during the r e- pro ductive per io d. Since L is a non-increa sing function, we see, that starting in a repro ductive p erio d ( p 3 > L ) we cannot obtain p 3 < L at the end o f this p erio d. This tells us that b efor e r epr o duction p erio d the stor age p erio d is imp ossible . If the c limate conditions (functions µ and ζ ) a re such that p 1 ( t ) = L ( t ) for a ll t ∈ [ t s , t 1 ] for some t s < t 1 , then according to (13) a plant can hav e the pe rio d with mixed control v 1 ∈ [0 , v ] for t ∈ [ t s , t 1 ]. Although this p ossibility cannot b e excluded in gener al, s uch mixed controls can arise o nly due to very sp ecific climate conditions and we do no t s eparate it as a sp ecial p erio d of plant life. If p 1 ( t ) − L ( t ) is incr easing fro m the left at t = t 1 , then one c an distinguish one more repr o ductive p erio d [ t 1 − s, t 1 ) for s ome s > 0. Throughout this pa pe r we follow the agreement to combine all suc h p erio ds together with stages with mixed c ontrols b et ween these p erio ds into one repro ductive p er io d. Let p 1 ( t ) − L ( t ) be decreas ing. Then for so me time int erv a l preceding to the repro ductiv e p erio d we have p 1 ( t ) > L ( t ) > p 3 ( t ) a nd therefor e on this time in ter- v al a plan t has a vegetative p erio d. W e call it perio d V . 1, in contrast to p erio d V . 2 character ized by relation p 1 ( t ) > p 3 ( t ) > L ( t ) (this distinction will b e useful for mono carpic pla n ts). There ar e 2 p ossibilities for the plant b ehavior b efore per io d V . 1: either it will have one more R -p erio d (if p 1 decreases low er than L ( t ) while it remains true that p 3 < L ( t )), or it will exist t 2 < t 1 : p 3 ( t 2 ) = L ( t 2 ). As men tioned befo re, we neglect the p oss ibilit y of mixed controls and consider the case ˙ p 3 ( t 2 ) < 0. In this case pe rio d V . 2 characterized by p 1 ( t ) > p 3 ( t ) > L ( t ) precedes the p erio d V . 1 . Although the a llo cation pat- 13 tern is the same in bo th p erio ds V . 1 and V . 2, the distinc- tion b etw een these per io ds is useful f or the study of phenol- ogy of mo no carpic plants. T o understand this difference let us co nsider the case, when the nonstr uctural car bo - hydrates cannot b e deconstructed due to external factors (i.e. ω ≡ 0, which implies, as was mentioned e arlier, that p 3 is no n-increasing) and the pr obability o f surviv al re- mains co nstant thr oughout the whole p erio d ( L ≡ const ). This implies that b efore p erio d V . 2 the repro duction p eri- o ds are not p oss ible ( p 3 > L ) and cons equent ly the plant exploits mono c arpic stra te gy . In the gener al case, when ω 6≡ 0 b oth p erio ds R and S ca n precede the V -pe rio d, or all the pr evious life of a plant can consist of one v egetative per io d. In the first case a plant po ssesses one more repro duction p erio d, which has bee n a lready ana lyzed. If b efore vegetativ e perio d there is no other pe rio d, then the plant is annual . Let now the S -per io d pr ecedes to the V -pe rio d. Then there exist t 4 , t 3 : t 4 < t 3 < t 2 , such that p 1 increases on [ t 4 , t 3 ] (due to the unfav orable climate conditio ns) and p 1 ( t 4 ) = p 3 ( t 4 ). W e sepa rate p erio d b etw een t 4 and t 3 in the season V . 2 . 1 ( p 1 > p 3 > L ( t ), but p 1 increasing), whic h distinctive feature is that although the climate c onditions ar e not c omfortable for photosynthesis a plant anyway al- lo c ates some p art of stor e d r esour c es to the c onstruction of the ve getative tissues, so as to c ome into the b etter c ondi- tions with a c ert ain amount of alr e ady develop e d ve getative mass . Now le t there exist some r : p 1 ( t ) < p 3 ( t ) for all t ∈ [ r , t 4 ). Then a plant enters a storage per io d. If the clima te conditions are unfavorable fo r all t < t 4 , that is, p 1 ( t ) < p 3 ( t ) for all t ∈ [0 , t 4 ), then the firs t per io d of time is only the storage of allo cated photosyn thate (this is hardly p ossible b ecause a seed has a p os sibilit y to stay this p erio d in do rmancy). If it is not the case , then there exist some moment s t 6 , t 5 , t 6 < t 5 < t 4 , such that p 1 is decreasing on [ t 6 , t 5 ] and p 1 ( t 6 ) = p 3 ( t 6 ). W e separate the p erio d ( t 5 , t 4 ), which we call p erio d S. 1 (when the clima te co nditions are dis adv antageous and all the allo ca ted material is sto red), a nd time-s pan ( t 6 , t 5 ) called p erio d S.2 (when the climate conditions are kindly , but all the allocated material is an y wa y stor ed f or a prepa- ration to the unfav o rable climate conditions). Both reproductive and v egetative perio ds can pre cede to the storage p erio d. It dep ends o n the clima te conditio ns and v alues of x 0 . It seems tha t in gener al we cannot s ay mo re ab out the time of propaga tion and type of the first p er io d. The r ea- son is that one ca n c ho ose the v alues of the initial pa rame- ters that are biolog ically inadequate and co nsequently ob- tain unr ealistic predic tions. F or e xample, if the climate conditions are chosen to be unfav orable for photosynthesis throughout all the time-interv al [0 , T ], then the mo del is inapplicable, beca use the strateg y to stay in dormancy all the per io d is not allowed in the mo del. T o exclude such biologically irrelev ant b ehavior, w e con- sidered in the pa per only the ca se, when the first p erio d after s prouting is vegetative. App endix 2: Pro of of the main prop o sition from Section 4 Pr o of. The problem (6), (7) can b e written in eq uiv alent form, using new v aria bles y i ( t ) := ax i ( t ), i = 1 , 2 , 3. Then we have: ˙ y 1 = v 1 ( t ) ag ( y 3 a ) − µ ( t ) y 1 , ˙ y 2 = ( v ( t ) − v 1 ( t )) ag ( y 3 a ) , ˙ y 3 = ζ ( t ) af ( y 1 a ) − v ( t ) ag ( y 3 a ) − ω ( t ) y 3 , y (0) = y 0 . (20) The corres po nding ma ximum problem is: max 0 ≤ v ( t ) ≤ 1 , 0 ≤ v 1 ( t ) ≤ v ( t ) , a ∈ [1 , ∞ ) Q a = ξ Z T t 0 L ( s ) ˙ y 2 ( s ) ds. (21) Now the pr oblem is similar to (1), (4), but with af ( y 1 a ) and ag ( y 3 a ) ins tead of f ( x 1 ) a nd g ( x 3 ). Using concavit y we have: f ( y 1 a ) = f ( 1 a y 1 + a − 1 a · 0) ≥ 1 a f ( y 1 ) + a − 1 a f (0 ) = 1 a f ( y 1 ). Thu s, for ev ery y 1 ( t ) ≥ 0, a ≥ 1 it holds af ( y 1 ( t ) a ) ≥ f ( y 1 ( t )) and therefore af ( y 1 ( t ) a ) and ag ( y 3 ( t ) a ) are non-decrea sing in a and sup a ∈ [1 , ∞ ) af ( y 1 ( t ) a ) and sup a ∈ [1 , ∞ ) ag ( y 3 ( t ) a ) y ields, when a → ∞ . Define the optimal tra jectories of the problem (6), (7) for a fixed a a s y ( · ). Now take arbitrary n > a and co nsider a sy stem ˙ y 1 = v 1 ( t ) ag ( y 3 a ) − µ ( t ) y 1 , ˙ y 2 = ( v ( t ) − v 1 ( t )) ag ( y 3 a ) , ˙ y 3 = ζ ( t ) nf ( y 1 n ) − v ( t ) ag ( y 3 a ) − ω ( t ) y 3 , y (0) = y 0 . (22) The solution of this system at time t sub ject to optimal- it y conditio n (20) we denote ˆ y ( t ). If ζ (0) > 0, then from nf ( y 1 ( t ) n ) > af ( y 1 ( t ) a ) we hav e that ˙ ˆ y 3 (0) > ˙ y 3 (0 , a ) and therefore there exis ts t ∗ > 0 : ˙ ˆ y 3 ( t ) > ˙ y 3 ( t, a ) ∀ t ∈ [0 , t ∗ ). Hence ˆ y 3 ( t ) > y 3 ( t, a ) and ag ( ˆ y 3 ( t ) a ) > ag ( y 3 ( t,a ) a ) for t ∈ (0 , t ∗ ). Let v and v 1 be the optimal controls for the system (20). There exist controls 0 ≤ ˆ v ≤ v , 0 ≤ ˆ v 1 ≤ v 1 for the system (22), such that ˆ v ( t ) ag ( ˆ y 3 ( t ) a ) = v ( t ) ag ( y 3 ( t,a ) a ) a nd ˆ v 1 ( t ) ag ( ˆ y 3 ( t ) a ) = v 1 ( t ) ag ( y 3 ( t,a ) a ). Consequently , ˆ y i ( t ) = y i ( t, a ), t ∈ [0 , t ∗ ), i = 1 , 2. 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