DYNAMICS OF A DISCRETE STOICHIOMETRIC TWO PREDATORS ONE PREY MODEL

Transcription

1 Journal of Biological Systems, Vol. 18, No. 3 (21) c World Scientific Publishing Company DOI: /S DYNAMICS OF A DISCRETE STOICHIOMETRIC TWO PREDATORS ONE PREY MODEL CONGBO XIE,,,MENGFAN, and WEI ZHAO,, School of Mathematics and Statistics Northeast Normal University, 5268 Renmin Street Changchun, Jilin, 1324, P. R. China Faculty of Science, Dalian Nationalities University Dalian, Liaoning, 1166, P. R. China Department of Engineering Mechanics Dalian University of Technology Dalian, Liaoning, 11624, P. R. China xiecb79@dlnu.edu.cn mfan@nenu.edu.cn zhaow@dlnu.edu.cn Received 18 August 29 Accepted 27 November 29 This paper derives a biologically reasonable discrete analog of a stoichiometric continuous two predators one prey model (i.e., LKEF model) proposed by Loladze et al. 1 By comparing the LKEF model to its discrete analog, we show that both the LKEF model and its discrete analogue exhibit similar dynamics but also some noticeable differences. In the discrete time setting, two predators exploiting a single prey can coexist not only at a stable equilibrium but also via oscillations, and the chaotic dynamics occur for biologically plausible parameter sets. Our results suggest that the stoichiometric mechanisms are robust to time discretization and the nutritional quality of prey has dramatic and counterintuitive impact on predator-prey dynamics, which agrees with the existing analysis of pelagic systems. Keywords: Ecological Stoichiometry; Competitive Exclusion Principle; Coexistence; Chaos; Lyapunov Exponents. 1. Introduction Predation and competition are among the most important interspecific interactions and contribute much to the structure and function of ecological communities. The competitive exclusion principle (CEP), sometimes referred to as Gause s Law, 2 states that two species competing for the exactly same resources can not stably coexist, or the maxim paraphrased as complete competitors cannot coexist. 3 The CEP, a theoretical concept that follows from rigorous mathematical modeling study, seems to be violated in many natural ecosystems. One of the best known examples is the paradox of the plankton or plankton paradox. 4 Various mechanisms have Corresponding author. 649

2 65 Xie et al. been proposed to explicate the coexistence of n species exploiting fewer than n resources, see Ref. 1 for a short review. The theory of ecological stoichiometry, 5 which provides rigorous and ubiquitous mechanisms for modeling population dynamics, has seen some exciting progresses in understanding and modeling ecosystems. 1, 5 9 Most existing classical models of population dynamics have been extensively examined assuming that all the trophic levels are chemically homogeneous and all the individuals are assumed to be made of a single constituent (usually referred to as energy). But in reality, any individual is composed of multiple chemical elements such as carbon (C), nitrogen (N), phosphorus (P), and sulphur (S), which together with oxygen (O) and hydrogen (H), comprise the bulk of biomass of all organisms. However, the concentrations of these key elements vary significantly among species. Theoretical and experimental studies based on the principles of ecological stoichiometry have shown that the nutritional 1, 5 9 quality of the prey can assert significant effect on the predator-prey dynamics. Loladze et al. 1 proposed a stoichiometric model (for simplicity, we refer to it as LKEF model) to analyze the competition between two predators exploiting one autotrophic prey. Their detailed theoretical and numerical studies reveal that the principles of ecological stoichiometry can be new mechanisms for coexistence and shed new light on trophic dynamics. It is widely recognized that the selection of time scale is very important in the study of biology and ecology. 1 Note that the LKEF model is defined on continuous time scale while the experimental data are usually discretely collected, many producers such as annual plants have nonoverlapping generations, and many herbivores have seasonal dynamics. Such scenarios call for models defined on discrete time scales. In mathematics, the discrete models can provide efficient computational models for numerical simulation. This also raises the question about the sensitivity of the main finding on LKEF model to the discretization of time. Fan et al. 6 and Sui et al. 9 compared the continuous stoichiometric LKE producer-grazer model 8 and KHE plant-herbivore model 7 to their discrete analogues, respectively. They systematically discussed the similarities and differences of those models. However, either LKE model or KHE model is governed by two-dimensional autonomous ODEs, which mathematically preclude any possibility of chaos while the LKEF model is three-dimensional and is essentially different from two-dimensional models mathematically. Motivated by these considerations, in this paper, we compare the dynamics of LKEF model with its discrete analogue and devote to finding out whether the following most important feature exhibited in the continuous LKEF model is robust to the time discretization: is it at all possible for two predators to stably coexist on one single prey of variable quality on discrete time scale? In the next two sections, we construct a discrete analogue of LKEF model and explore its dynamics qualitatively. With the help of ecological matrix, it is shown that two predators exploiting a single prey can coexist. In Sec. 4, we numerically study the discrete analogue of LKEF model with Monod functional response and systematically expound the similarities and differences between LKEF model and its discrete analogue.

3 Dynamics of a Discrete Stoichiometric Two Predators One Prey Model A Discrete Analogue of (2.1) Consider the following LKEF stoichiometric model depicting the dynamics of two predators competing for a single prey 1 ( dx dt = rx 1 dy 1 dt = e 1 min dy 2 dt = e 2 min x min(k, (P s 1 y 1 s 2 y 2 )/q) ( 1, (P s 1y 1 s 2 y 2 )/x ( s 1 1, (P s 1y 1 s 2 y 2 )/x s 2 ) f 1 (x)y 1 f 2 (x)y 2, ) f 1 (x)y 1 d 1 y 1, ) f 2 (x)y 2 d 2 y 2, (2.1) where x, y 1 and y 2 are the densities of the prey (assumed to be a photoautotroph such as phytoplanktonic algae) and the two predators, respectively. The detailed derivation of (2.1) isref.1. For convenience, the parameters and their biological meanings are listed in Table 1. Now, we are at the right position to derive a discrete analogue of the LKEF model (2.1). There are at least three ways or procedures to discretize the continuous models known to theoretical ecologists. 11, 12 The first approach is based on direct discretization of the derivative in continuous models. The second employs the differential equations with piecewise constant arguments by assuming that the per capita growth rate stays constant during the interval of time [t, t+1]. The third derivation is based on integrating the model one time step forward from t to t +1. Compared with the other two approaches, the second, which is employed in Ref. 6, Table 1. Parameters (P) of (2.1) and (2.2) with their values (V) from 1 used for numerical simulations. P Biological definition V 1 V 2 Units r Intrinsic growth rate of the prey day 1 K Resource carrying capacity determined by light (mg C)/l e 1 Maximal conversion rate of the 1st predator.72.8 e 2 Maximal conversion rate of the 2st predator d 1 Loss rate of the 1st predator day 1 d 2 Loss rate of the 2st predator.2.1 day 1 s 1 Constant P:C of the 1st predator (mg P)/(mg C) s 2 Constant P:C of the 2st predator.5.38 (mg P)/(mg C) q Minimal possible P:C of the prey.4.3 (mg P)/(mg C) P Total P in the system.3.36 (mg P)/l f 1 Ingestion rate of 1st predator f 2 Ingestion rate of 2nd predator a 1 Half-saturation constant of the 1st predator.3.36 (mg C)/l a 2 Half-saturation constant of the 2nd predator.2.45 (mg C)/l c 1 Maximal ingestion rate of the 1st predator.7.6 day 1 c 2 Maximal ingestion rate of the 2nd predator.8.63 day 1

4 652 Xie et al. is better and is widely used to discretize continuous model of population dynamics 11, 12 in most cases. Following the method and procedure used in Ref. 6 (the second approach mentioned above), we reach the following discrete stoichiometric two predators one prey model { ( ) x(n) x(n +1)=x(n)exp r 1 min(k, (P s 1 y 1 (n) s 2 y 2 (n))/q) } 2 f i (x(n))y i (n), x(n) i=1 y 1 (n +1) { ( = y 1 (n)exp e 1 min y 2 (n +1) { ( = y 2 (n)exp e 2 min 1, (P s 1y 1 (n) s 2 y 2 (n))/x(n) s 1 1, (P s 1y 1 (n) s 2 y 2 (n))/x(n) s 2 ) f 1 (x(n)) d 1 }, ) f 2 (x(n)) d 2 }, (2.2) which is a discrete time analogue of (2.1). The time scale of (2.2) is the set of integers Z, i.e., n Z, where, without any loss of generality, the step size is standardized to be 1. Two sample sets of parameters in (2.2) used for numerical simulations are listed in Table 1. In (2.2), throughout this paper, we assume that f i (x),i =1, 2 are bounded smooth functions satisfying f i (x) =xp i (x), f i () =, f i () <, f i Then, it follows that, for any x>, (x) >, f i (x) <, x >. (2.3) lim p i(x) =f i () <, x p i (x) <. It is easy to see that the nonnegative quadrant of R 3 is positively invariant with respect to (2.2). Considering the biological significance of the model, we always assume that x() >,y 1 () >, and y 2 () > in the following discussion. 3. Dynamics of (2.2) It is not difficult to show that any solution of (2.2) is bounded above by some positive constants depending on the initial population size and the parameter values (Appendix A). Moreover, the set defined by (A.2) is positively invariant with respect to (2.2) and is globally attractive for (2.2). Appendix B shows that (2.2) always has two axial or boundary equilibria E =(,, ) and E k =(min(k, P/q),, ) = (k,, ). For general function f i (x)

5 Dynamics of a Discrete Stoichiometric Two Predators One Prey Model 653 satisfying (2.3), there can be several other boundary equilibria and positive equilibria. With the help of the so-called Jury test, 13 one can easily show that E is unstable and is always a saddle, which implies that the total extinction of three species is impossible for (2.2). In addition, E k is locally asymptotically stable if <r<2and e i min(1,p/s i k)f i (k) <d i while it is unstable if r>2ore i min(1,p/s i k)f i (k) >d i. That is to say, if both predators have high death rate and the growth rate of the prey is small, then the predators y 1 and y 2 become extinct while the prey x survives at its carrying capacity. (2.2) can have some internal or positive steady state, say E (x,y1,y 2 ), i.e., the predators y 1 and y 2 can coexist on a single prey x at E.Fortheinternal equilibrium E, its stability is not easy to deal with. One can determine the type of species interactions at E by investigating the ecosystem matrix M(x,y1,y 2) of (2.2) defined by (C.2) in Appendix C. By (B.3) in Appendix B, the ecosystem matrix M(x,y1,y 2) can be classified into three different types +/ +/ +/ (M g ) +, (M b ), (M m ) +, + where +, and denote positive or beneficial effect, negative or harmful effect and no effect, respectively. Now we are at the right position to investigate the effect of the prey quality on the species interactions and coexistence. For simplicity, as in Ref. 1, wecallthe predator with higher P demand (i.e., with a higher P : C ratio, s i )asp-richpredator and the predator with lower P demand as P-poor predator. Based on the ecosystem matrix M(x,y1,y 2 ), our discussion is divided into three cases. Case 1. Prey quality is good for both predators, i.e., (P s 1 y 1 s 2 y 2 )/x > s i. Now, the ecosystem matrix M takes the form (M g ). The interactions between the predators and the prey are both (+, ), the conventional predator-prey interactions. By (C.1), the component x of E should simultaneously satisfy f 1 (x )= d 1, f 2 (x )= d 2, e 1 e 2 which is almost impossible since the set of parameter values satisfying both equations is of measure zero. Therefore, E can hardly exist, in other words, the CEP holds and the predators cannot coexist on the single prey. Case 2. Prey quality is bad for both predators, i.e., (P s 1 y 1 s 2 y 2 )/x < s i. In this case, the ecosystem matrix M takes the form (M b ), which shows that both the predators and the prey compete with each other and the competition lies in three species not only internally but also interspecificly for the limited P content, and the interaction between the predator and its prey is changed from the

6 654 Xie et al. traditional (+, ) to(, ). The predators and the prey can coexist at the positive equilibrium E. In fact, (2.2) now reduces to { ( ) x(n) x(n +1)=x(n)exp r 1 min(k, (P s 1 y 1 (n) s 2 y 2 (n))/q) } 2 p i (x(n))y i (n), i=1 { (P s 1 y 1 (n) s 2 y 2 (n)) y 1 (n +1)=y 1 (n)exp e 1 p 1 (x(n)) d 1 s 1 { (P s 1 y 1 (n) s 2 y 2 (n)) y 2 (n +1)=y 2 (n)exp e 2 p 2 (x(n)) d 2 s 2 }, }. (3.1) By the predators equations in (3.1), the components of E satisfy d 1 s 1 p 1 (x = d 2s 2 )e 1 p 2 (x = P s 1 y1 )e s 2y2 := A, (3.2) 2 which determines the value of x.byf (x,y 1,y 2 )=, ( x ) r 1 = min{k, A/q} 2 p i (x )yi i=1 := B, (3.3) then the values of y i are given by y 1 = p 2(x )A + s 2 B p 2 (x )P p 1 (x )s 2 p 2 (x )s 1, y 2 = p 1(x )A s 1 B + p 1 (x )P p 1 (x )s 2 p 2 (x )s 1. (3.4) Note that, from (3.2), x is independent of K while y1 is increasing with K and y2 is decreasing with K since B, defined by (3.3), monotonically increases with K. Here, we can assume that p 1 (x )s 2 p 2 (x )s 1 >. That is to say, increasing the energy flow into this food web (i.e., increasing K) does not change the prey s density, but can benefit the P-poor predator y 1 and hurt the P-rich predator y 2 since (2.2) is of ecological stoichiometry type and the food web is determined by both the quantity and the quality of the prey. This will be shown more clearly by the numerical analysis in the next section. Case 3. Prey quality is good for one predator but bad for the other, say s 1 < (P s 1 y 1 s 2 y 2 )/x < s 2.

7 Dynamics of a Discrete Stoichiometric Two Predators One Prey Model 655 The ecosystem matrix M reads (M m ), where the predator y 1 is in predator-prey relationship with the prey ((+, )), while the predator y 2 is competing with the prey ((, )). In this case, (2.2) reads { ( ) x(n) x(n +1)=x(n)exp r 1 min(k, (P s 1 y 1 (n) s 2 y 2 (n))/q) } 2 p i (x(n))y i (n), (3.5) i=1 y 1 (n +1)=y 1 (n)exp{e 1 x(n)p 1 (x(n)) d 1 }, { y 2 (n +1)=y 2 (n)exp e 2 (P s 1 y 1 (n) s 2 y 2 (n)) s 2 p 2 (x(n)) d 2 then the x-component of E satisfies x d 1 = e 1 p 1 (x ), and does not change with K. In addition, yi are also given by (3.4). Similar to Case 2, one can draw the conclusion of the coexistence of two predators on one prey. In this case, these three species can coexist at the positive equilibrium E or even via oscillations (see numerical simulations in Sec. 4). The above analysis based on the ecosystem matrix shows that the nutritional prey quality (changes in the prey s stoichiometry) has dramatic and counterintuitive impact on the dynamics of the food web interactions. The two predators can coexist on one prey if the prey quality is bad for at least one of the two predators. }, 4. Numerical Analysis and Discussion In this section, we carry out systematic numerical analysis for (2.2) to confirm and complement our analytical analysis in the previous section. Particularly, we will explore the coexistence of all three species. For simplicity and without any loss of generality, we choose the Monod type function f i (x) =c i x/(a i + x) asthe functional response of the predators. The parameter values listed in Table 1 are adapted from Ref. 1 and, in fact, the biologically realistic values in V 1 are originally from Refs In the following numerical simulations, the initial values are set to x() =.5, y 1 () =.25, y 2 () =.25. First, we focus on the first set of the parameter values V 1. In this case, the 2nd predator y 2 needs more P (s 2 >s 1 ) and has a higher growth rate (e 2 f 2 (x(n)) d 2 > e 1 f 1 (x(n)) d 1 ) if prey quality is good. The principle aim here is to show that all three species can coexist at a positive equilibrium, which is asymptotically stable as well as structurally stable. We are interested in the dependence of coexistence on energy flow into the system (light intensity) since light intensity can change prey quality. 16

8 656 Xie et al. (a) (b) (c) (d) (e) (f) Fig. 1. Bifurcation diagram of the species densities plotted against K for the discrete LKEF model (2.2) (a c) and the continuous LKEF model (2.1) (d f). The values of other parameters are given as V 1 in Table 1. Figure 1 shows a bifurcation diagram of the species densities plotted against K. The values of the parameters are given by V 1 in Table 1 except that K varies from 2. One can easily see that the bifurcation diagram of (2.2) infig.1 is similar to that of the continuous LKEF model (2.1), which has been systematically studied in Ref. 1. In order to be self-contained, we will address a detailed explanation of the bifurcation diagram of (2.2) infig.1. For <K<.1, neither of the predators survives due to the very low food quantity and starvation. For.1 <K<.92, the CEP works and y 2 precludes y 1 and coexists with the prey due to the fact that y 2 has a higher growth rate. For.1 <K<.3,

9 Dynamics of a Discrete Stoichiometric Two Predators One Prey Model 657 the increase of K leads to the increase of the density of the 2nd predator without changing the prey s density at the equilibrium. The prey has no benefit of the enrichment. When K =.3, (2.2) experiences the so-called Rosenzweig s paradox of enrichment. Further enrichment (.3 <K<.92) leads to the destabilization of the equilibrium via Hopf bifurcations. The 2nd predator and the prey coexist at some oscillation for.3 <K<.92 and the period of the oscillation varies with K, e.g. in Fig. 3, the attractor is a limit cycle with different period, i.e., 62 and 48 in (a) and (b), respectively. From Fig. 1, one can find that, for.3 <K<.92, the dynamics of (2.2) seems chaotic. In Fig. 2, the zoomed picture of Fig. 1, there are many sparse zones with different width. It is natural to ask whether (2.2) admits chaotic attractors. Our Lyapunov exponent studies in Fig. 4 show that the Lyapunov exponents of (2.2) λ i (i =1, 2, 3) are all nonpositive with K varying from 2 and the sign spectrum of the Lyapunov exponents is (,, ) or(,, ), which means that the attractor is a periodic orbit (limit cycle) or quasi periodic (a) (b) Fig. 2. Zoomed picture of Fig. 1 with K varying from (a) The prey x, (b) The 2nd predator y 2.

10 658 Xie et al..8.6 x y 1 y time (a) k = x y 1 y time (b) k =.8 Fig. 3. Attractors of (2.2). (a) K =.6, the attractor is a positive periodic solution of period 62. (b) K =.8, the attractor is a positive periodic solution of period 48. The values of other parameters are given as V 1 in Table 1. orbit (torus orbit). 17 Therefore, if the parameters take the values in V 1,(2.2) does not admit any chaotic behavior although it seems chaotic from the bifurcation diagrams. Note that the quality of the prey decreases with the increase of K. As qualitative analysis in Sec. 3 suggests, for.92 <K<1.8, the numerical simulations (Fig. 1) confirm that both predators exploiting a single prey can coexist at a stable positive equilibrium. With the increase of energy flow K (.92 < K < 1.8), the equilibrium of the prey remains constant and the density of y 2 (the P-rich predator) decreases. But the density of y 1 (the P-poor predator) increases and hence can successfully invade into the system. Both y 1 and y 2 are limited by the P content of the prey, i.e., the quality of the prey. The food quality worsens gradually although the food quantity is abundant and, eventually, the P-rich predator y 2 is extinct at K =1.8 due to very low quality of the prey.

11 Dynamics of a Discrete Stoichiometric Two Predators One Prey Model 659 λ i (i=1, 2, 3) 1 λ 1 λ 2 λ K Fig. 4. Spectra of Lyapunov exponents for three species plotted against K for (2.2) with parameters V 1. As K increases from , the density of the P-poor predator y 1 decreases and tends to zero due to the worsening prey quality. In this case, two predators cannot coexist on the single prey. Figure 5 depicts the bifurcation diagram of (2.2) with the parameters taking values indicated by V 2 listed in Table 1. One can give a detailed explanation of diagram as in Fig. 1. In this case, two predators exploiting a single prey can coexist not only at a stable equilibrium (.3 <K<.615) but also via periodic oscillations (.615 <K<.8). Neither y 1 nor y 2 can survive due to the very low abundance of the prey for <K<.1 and the 2nd predator (P-rich predator) cannot persist for K>.8 due to the very low prey P : C ratio, i.e., very low prey quality. In fact, our extensive numerical analysis shows that the P-poor predator y 1 goes to extinction when K is large enough (K >5.45). In Figs. 1 and 5, the coexistence of two predators on a single prey corresponds to the ecosystem matrix taking form of (M b )and(m m ), respectively. In fact, let s := (P s 1 y 1 s 2 y 2 )/x, Fig.6 gives the picture of s with respect to K when the parameters in (2.2) take values of V 1 (Fig. 6a) and V 2 (Fig. 6b), respectively. When.92 <K<1.8, one has s<s 1 <s 2 in Fig. 6a, which means that the prey quality is bad for both predators, and Fig. 1 shows that the two predators coexist on a single prey. In Fig. 6(b), when.35 <K<.615, one has s 1 <s<s 2,which

12 66 Xie et al. (a) (b) (c) (d) (e) (f) Fig. 5. Bifurcation diagram of the species densities plotted against K for the discrete LKEF model (2.2) (a c) and the continuous LKEF model (2.1) (d f). The values of other parameters are given as V 2 in Table 1. means that the prey quality is bad for predator y 2 but good for predator y 1,and the two predators coexist on a single prey (Fig. 5). To conclude, in this paper, we systematically explore the dynamics of the discrete LKEF model (2.2) both analytically and numerically. Our main finding is that two predators exploiting a single prey can coexist not only at a stable equilibrium but also via oscillations. Our studies also show that the poor prey quality is detrimental to the predators growth and can even drive the predators to extinction. The reason is that the prey with very low P : C ratio, despite being a good source of calories (abundance in quantity), does not provide enough P to support

13 Dynamics of a Discrete Stoichiometric Two Predators One Prey Model 661 (a) (b) Fig. 6. Prey quality ((P s 1 y 1 s 2 y 2 )/x := s) against K. (a) the parameters in (2.2) take values of V 1 and, when.92 <K<1.8, s<s 1 <s 2, the prey quality is bad for both predators. (b) the parameters in (2.2) take values of V 2 and, when.35 <K<.615, s 1 <s<s 2, the prey quality is bad for predator y 2 but good for predator y 1. the predators growth. The low-quality of the prey can halt oscillations and stabilize the system at an equilibrium or the oscillations precede stabilization to steady state (Fig. 5), and any further enrichment benefits the prey but leads to further decline in the density of the predators, a phenomena known as the paradox of energy enrichment. 8 When the prey quality is good (i.e., P : C is high), these models behave like traditional energy-based predator-prey models such as Lotka-Volterra type models. The stoichiometric effects of the prey quality on predators are robust to the discretization of time as shown by Refs. 6 and 9. While stoichiometric effects of low prey quality are present in both (2.1) and (2.2), there are quantitative and qualitative effects that arise from discretization. The numerical simulations in Figs. 1 and 5 reveal that both the LKEF model and its discrete analogue exhibit similar dynamics. For example, both the discrete and the continuous LKEF model, i.e., (2.2) and(2.1), have almost the same coexistence range of parameter K: e.g.k (.92, 1.8) in Fig. 1 and

14 662 Xie et al. y y y x y x.3.4 (a) (b) Fig. 7. Comparison of the orbit in phase space. (a) An orbit of (2.2). (b) An orbit of (2.1). Here K =.485 and other parameters take the values defined by V 1 in Table 1. K (.35,.8) in Fig. 5; whenk is sufficiently large, (2.2) and(2.1) have similar qualitative dynamics: e.g. K>1.8 in Fig. 1 and K>.83 in Fig. 5. But there are also some noticeable differences. The amplitudes of the species densities in the discrete LKEF model are larger than that of the continuous (see Figs. 1, 5 and 7). In addition, the boundary equilibrium of the discrete LKEF model (2.2) loses its stability much earlier than that of (2.1) and the limit cycle dynamics are presented in a larger range of parameter values of K (see Figs. 1 and 5). Moreover, the discretization of time can significantly affect the nature of the attractors. For example, the attractor of the continuous LKEF model (2.1) is either a boundary equilibrium or an internal equilibrium or a limit cycle (see (d f) in Figs. 1 and 5) while the attractor of the corresponding discrete model (2.2) can be a boundary equilibrium, an internal equilibrium, a limit cycle, a quasi periodic cycle, or even a strange attractor (chaos) (see Fig. 8). Figure 8 gives the bifurcation diagram of (2.2) with respect to the parameter r and demonstrates the perioddoubling route to chaos. Figure 9 also supports this claim since the maximum Lyapunov exponent λ m > whenr> Therefore, the chaotic dynamics and a strange attractor can emerge in (2.2) for biologically plausible parameter values. This further supports Andersen s argument 14 that the stoichiometric systems can admit chaotic dynamics.

15 Dynamics of a Discrete Stoichiometric Two Predators One Prey Model 663 Fig. 8. Bifurcation diagram of the species densities plotted against r for the discrete LKEF model (2.2). Here K =1.5, r varies from to 3.5 and the values of other parameters are given as V 1 in Table λ m r Fig. 9. Spectra of the maximum Lyapunov exponent(mle) of (2.2) against r corresponding to Fig. 8. λ m > whenr>2.875, which shows the existence of chaos.

16 664 Xie et al. Acknowledgements We thank the anonymous referees for their careful readings and many suggestive comments, which greatly improve the presentation of the manuscript. This research has been supported partially by NSFC , NSFC and NCET (MF), and DLNU Youth Research Fund project (28A28) (CBX). References 1. Loladze I, Kuang Y, Elser JJ, Fagan WF, Coexistence of two predators on one prey mediated by stoichiometry, Theor Popul Biol 65:1 15, Gause GF, The Struggle for Existence, Williams & Wilkins, Baltimore, MD Hardin G, Competitive exclusion principle, Science 131: , Hutchinson GE, The paradox of the plankton, American Naturalist 95: , Sterner RW, Elser JJ, Ecological Stoichiometry, Princeton University, Princeton, NJ, Fan M, Loladze I, Kuang Y, Elser JJ, Dynamics of a stoichiometric discrete producergrazer model, J Differ Equ Appl 11: , Kuang Y, Huisman J, Elser JJ, Stoichiometric plant-herbivore models and their interpretation, Math Biosc Eng 1: , Loladze I, Kuang Y, Elser JJ, Stoichiometry in producer-grazer systems: linking energy flow with element cycling, Bull Math Biol 62: , Sui GY, Fan M, Loladze I, Kuang Y, The dynamics of a stoichiometric plant-herbivore model and its discrete analog, Math Biosc Eng 4:29 46, Frankham R, Brook BW, The importance of time scale in conservation biology and ecology, Ann Zool Fennici 41: , Gurney WSC, Nisbet RM, Ecological Dynamics, Oxford University Press, New York, Turchin P, Complex Population Dynamics: A Theoretical/Empirical Synthesis, Princeton University Press, Jury EI, The inners approach to some problems of system theory, IEEE Trans Automatic Contr AC-16:233 24, Andersen T, Pelagic Nutrient Cycles: Herbivores as Sources and Sinks, Springer- Verlag, New York, Elser JJ, Urabe J, The stoichiometry of consumer-driven nutrient recycling: theory, observations, and consequences, Ecology 8: , Urabe J, Sterner RW, Regulation of herbivore growth by the balance of light and nutrients, PNatlAcadSci93: , Ahmed O, Mohamed Z, Nejib S, Control of chaos through an instantaneous Lyapunov exponent targeting control algorithm, Int J Bifurcat Chaos 18: , 28. Appendix A: Boundedness In this section, we present some preliminary results for (2.2) such as the boundedness and positive invariance of the solutions of (2.2).

17 Dynamics of a Discrete Stoichiometric Two Predators One Prey Model 665 Theorem Appendix A.1 For (2.2), the following claims hold: for any n Z +, <x(n) max{x(), (ke (r 1) )/r} U, <y 1 (n) max{y 1 (),v 1 } exp{2e 1 f 1 (U) 2d 1 } V 1, <y 2 (n) max{y 2 (),v 2 } exp{2e 2 f 2 (U) 2d 2 } V 2. (A.1) where k := min{k, P/q} and v i is any number satisfying e i f i (Ue r pi(u)vi ) <d. The proof is similar to those in Theorems 3.1 and and Appendix C, 8 so the details are omitted here. Let {(x, y 1,y 2 ):<x<(ke (r 1) )/r, <y 1 <v 1, <y 2 <v 2 }. (A.2) Theorem Appendix A.1 implies that is positively invariant with respect to (2.2). Indeed,wehavethat is globally attractive for (2.2). Appendix B: Boundary Equilibria This section explores the existence and stability of boundary equilibria of (2.2). In order to facilitate the discussion below, we rewrite (2.2) in the following form: x(n +1)=x(n)exp{F (x(n),y 1 (n),y 2 (n))}, y 1 (n +1)=y 1 (n)exp{g 1 (x(n),y 1 (n),y 2 (n))}, (B.1) where y 2 (n +1)=y 2 (n)exp{g 2 (x(n),y 1 (n),y 2 (n))}. ( ) x(n) F (x(n),y 1 (n),y 2 (n)) = r 1 min(k, (P s 1 y 1 (n) s 2 y 2 (n))/q) 2 p i (x(n))y i (n), i=1 ( G i (x(n),y 1 (n),y 2 (n)) = e i min 1, (P s ) 1y 1 (n) s 2 y 2 (n))/x(n) s i x(n)p i (x(n)) d i. After some algebraic calculations, the Jacobian of (2.2) reads e F + xe F F xe F F xe F F y 1 y 2 J(x, y 1,y 2 )= y 1 e G G1 1 e G1 + y 1 e G G1 1 y 1 e G G1 1 y 1 y 2 y 2 e G G2 2 y 2 e G G2 2 e G2 + y 2 e G G2 2 y 1 y 2 (B.2)

18 666 Xie et al. where F = r min(k, ((P s 1 y 1 s 2 y 2 )/q)) F y i = G i = G i y j = 2 p i(x)y i, p i (x) <, K < (P s 1 y 1 s 2 y 2 )/q, rqs i x (P s 1 y 1 s 2 y 2 ) 2 p i(x) <, K > (P s 1 y 1 s 2 y 2 )/q, e i (p i (x)+xp i (x)) >, (P s 1y 1 s 2 y 2 )/x > s i, (P s 1 y 1 s 2 y 2 ) e i p i s (x) <, (P s 1y 1 s 2 y 2 )/x < s i, i, (P s 1 y 1 s 2 y 2 )/x > s i, i=1 s i e i p i (x) <, (P s 1 y 1 s 2 y 2 )/x < s i. s j The equilibrium (x,y1,y 2)of(2.2) solves the algebraic equations x = x exp{f (x,y1,y 2 )}, y1 = y 1 exp{g 1(x,y1,y 2 )}, y2 = y 2 exp{g 2(x,y1,y 2 )}. (B.3) It is trivial to show that (2.2) has only two axial or boundary equilibria E =(,, ) and E k =(min(k, P/q),, ) = (k,, ). For general function f i (x) satisfying (2.3), there can be several other boundary equilibria and positive equilibria. The stability of these equilibria can be routinely studied via the so-called Jury test 13 when some specific functions f i (x) aregiven. For the stability of the axial equilibria of (2.2), we calculate the Jacobian of (2.2) at E and E k, J(E )= e r e d1 e d2, J(E k )= One can easily reach the following conclusion. 1 r k F/ y 1 k F/ y 2 e G1 e G2 e G2 Theorem Appendix B.1 For (2.2), E is always a saddle. E k is locally asymptotically stable if it is unstable if <r<2 and e i min(1,p/s i k)f i (k) <d i ; r>2 or e i min(1,p/s i k)f i (k) >d i..

19 Dynamics of a Discrete Stoichiometric Two Predators One Prey Model 667 Appendix C: Internal Equilibria and Ecosystem Matrix In this section, we focuses on the internal equilibria (2.2). If the system of algebraic equations F (x, y 1,y 2 )=G 1 (x, y 1,y 2 )=G 2 (x, y 1,y 2 )=, x >, y i >. (C.1) has a solution, then (2.2) has an internal or positive equilibrium, say E (x,y1,y 2). The stability of E is not easy to deal with. In order to determine the type of species interactions at E, we introduce and investigate the ecosystem matrix of (2.2). The Jacobian of (2.2) ate reads Define J(E )= 1+x F y1 G 1 y2 G 2 x F y 1 x F y 2 1+y1 G 1 y G 1 1 y 1 y 2 y2 G 2 1+y G 2 2 y 1 y 2 1 x = 1 + y1 1 y2 M(x,y1,y 2):= F G 1 G 2 F y 1 F G 1 G 2 F y 2 G 1 y 1 G 1 y 2 G 2 y 1 G 2 y 2 F y 1 F y 2 G 1 y 1 G 1 y 2 G 2 y 1 G 2 y 2. (C.2) as the ecosystem matrix of (2.2) ate,wheretheijth term measures the effect of the jth species on the per capita growth rate of the ith species. The ecosystem matrix M(x,y 1,y 2) can help to determine the type of species interactions at E of (2.2) (see Sec. 3 for more details).