Multistability in an open recruitment food web model

Applied Mathematics and Computation 163 (2005) 275–294 www.elsevier.com/locate/amc

Multistability in an open recruitment food web model Ismael Velazquez a, David Kaplan b,c, Jorge X. Velasco-Hernandez d,*, Sergio A. Navarrete

b

a

Departamento de Matem
aticas, Universidad Auton
oma Metropolitana-Iztapalapa, Apdo-Postal 55-534, M
exico D.F. 09340, Mexico b Departamento de Ecolog
ıa, Estaci
on Costera De Investigaciones Marinas and Center for Advanced Studies in Ecology and Biodiversity, Pontificia Universidad Cat
olica de Chile Casilla 114-D, Santiago, Chile c Department of Wildlife, Fish and Conservation Biology, University of California, Davis, CA 95616, USA d Programa de Matem
aticas Aplicadas y Computaci
on, Instituto Mexicano del Petr
oleo, Eje Central L
azaro C
ardenas 152, M
exico D.F. 07730, Mexico

Abstract A general model for three species food web found in marine systems is analyzed. Following single-species approaches to model the dynamics of marine organisms, we relaxed the assumption of closed population structure and examine the behavior of webs in which some or all the component populations have completely open dynamics, that is, there is no connection between the arrival of new individuals and local reproductive output. Recent reviews have shown that coexistence among species with self-recruitment with those with completely open-recruitment is the norm in marine habitats. As part of a study of tri-trophic food web models with combinations of self-versus open-recruitment, we describe here the stability properties of a food web with omnivory where the basal and the intermediate predator species have open populations and the top predator reproduces locally. This system can have a maximum of four critical points and at least one corresponding to the omnivore-free equilibrium point. In this case our system reduces to a two level food web without omnivory. This equilibrium point has stability properties that depend on the capacity of invasion of the omnivore species. If the omnivore succeeds in invading the community then a three level food web can be

*

Corresponding author. E-mail address: [email protected] (J.X. Velasco-Hernandez).

0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.02.005

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established but with more complex stability properties. When the equilibrium point without the top predator is unstable, then there may exist three more critical points, at least one of which is asymptotically stable. The exact number of critical points that may exist depends on the food web parameter space. We speculate that the restrictive conditions for three-species stability could explain the scarcity of this particular combination of dispersal abilities in natural communities. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Food web; Open recruitment; Mathematical model; Omnivory; Bistability

1. Introduction Traditional conceptions of food webs portray species as belonging to defined trophic levels, whose direct interactions with other species in the web are limited to those occurring between prey and predators, one trophic level above or below the focal species (e.g. [6,8,23,24,26]. Early empirical data gave some support to this food-chain view of communities [26]. Consequently, most theoretical studies constructed simplified tri-trophic food-web modules with chain-like predator–prey interactions, sometimes involving more than one competing prey species (e.g. [9,10,12,13,26,33]. Empirical studies over the past few decades have shown, however, that species in most natural food webs cannot be neatly classified into static trophic levels; they usually are reticulately connected to various species at different levels of the food web [4,17,20,22,29,31,41]. Indeed, omnivory, feeding on more than one trophic level, seems to be the dominant feature of many marine, aquatic and terrestrial communities (e.g. [1,4,5,19,25,30,32,35,36,38,40]. Following this reticulate, and probably more realistic, vision of food webs, theoretical studies have started to explicitly incorporate omnivory and intraguild predation (IGP, sensu [31] when investigating the dynamical properties of food web regulation [14,15,28]. Theoretical investigation has followed two different routes. On the one hand, research has been directed at understanding the conditions under which one would expect stable coexistence of all species involved in direct interactions within simplified tri-trophic food webs [11,27,28]. On the other hand, the approach has been to start with slightly more complex formulations of the interactions among species (e.g. including saturating functional responses), producing food web modules that are far from stable equilibrium (e.g. exhibiting chaos), and then examining if the inclusion of omnivorous links tends to stabilize, or at least bound the system oscillations further from extinction [14]. These two complementary approaches have shown that the dynamics and stability conditions of subweb modules with and without omnivory are indeed very different, highlighting the contrast between the food-chain and the reticulate views of food webs. Conclusions about the

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actual effect of omnivory on food web stability depend, however, on how the system is modeled. Omnivory tends to destabilize food webs that otherwise are stable, making conditions for coexistence much more stringent [11]. In contrast, omnivory tends to stabilize food webs that are far from equilibrium in the first place, either eliminating chaotic behavior altogether or generating interior attractors for systems without positive solutions [14]. The common theme of both lines of research has been to visualize the component species in the food web as having closed population structure within the spatial scale implicit in the models. By ‘‘closed’’ we mean that immigration and emigration can be safely omitted and that population growth rate is strictly dependent on the local reproduction of adults. This assumption is at odds with the essentially open nature of populations of marine species. Most marine species posses a larval stage that must spend from weeks to months in the water column before settling back in the adult habitat, usually far away from the parent population [34,42]). A number of theoretical models that suit marine organisms with open recruitment have been developed to examine the dynamics single species (e.g. [34]; Roughgarden and Iwasa, 1986) as well as interactions between two species (e.g. [3,7,43]. The logical extension of this work is to examine the dynamics of tri-trophic food web models with open recruitment. However, an important attribute of marine communities is the coexistence of species with widely differing dispersal strategies, from species that lack a larval stage at all, and therefore only reproduce locally, to those that have long lived larvae spending several months in the water column [19,37,39]. Indeed, recent reviews show that coexistence of species with self- and openrecruitment is the norm in marine systems ([44,45]). Therefore, at most spatial scales of analysis there will be species that reproduce locally interacting with species whose dynamics is largely given by recruitment from distant populations (e.g. [19,46]. As part of the study of tri-trophic food web models with combinations of self-versus open-recruitment, we describe here the stability properties of tri-trophic food webs with omnivory where the basal and the intermediate predator species have open populations and the top predator reproduces locally. Although this is only one of the combinations of life histories observed in natural communities, and one that appears to be relatively uncommon (see Section 5), we show below that its stability properties are particularly interesting to analyze. Although inspiration for the study comes from marine food webs, the models apply to any food web in which local dynamics is dominated by immigration and the reproductive output of some species is exported beyond the boundaries of the system. The paper is organized as follows: in Section 2 we present the general food web model with combination of open and closed populations; in Section 3 we show the existence of equilibria; next in Section 4 we do the stability analysis of equilibrium points; in Section 5 we present some numerical results and the last Section contains our conclusions.

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2. Three-species open food web The model consists of the following set of differential equations that represent the resource species R the intermediate species N and the top predator species P that in this model is the omnivore species that has local reproduction. The model is the representation of a simple food web where recruitment of new individuals in two lower levels is non-local in the sense that recruitment is independent of the population density in each level. The equations are: dR ¼ K1  a12 RN  a13 RP  m1 R; dt dN ¼ K2  a23 NP  m2 N ; dt

ð1Þ

dP ¼ a31 PR þ a32 PN  m3 P ; dt where Ki is the recruitment rate of species i with i ¼ 1 representing the resource species and i ¼ 2 the intermediate species; mi is the mortality rates of each species labeled in ascending order in the food web, and aij are the predation rates of species j on species i, when j < i; but it is a conversion rate of prey into predator numbers if j > i. The food web is a modification of the classical Lotka–Volterra system but modified to account for non-local recruitment and omnivory as indicated above, and as an extension of Gaines and Lafferty’s 7] models for two species interactions. 2.1. Existence of equilibria To analyze the food web the first task is to find the biologically feasible equilibrium points defined as those where the growth rate of all species is identically zero. This open food web presents a diversity of equilibria with stability properties depending upon parameter values as will be shown in the rest of the paper. To start we have the following result on feasible equilibrium points defined as those with non-negative equilibrium densities for each species: Theorem 1. The open three species food web (1) can have up to four and at least one biologically feasible
equilibria. The equilibrium points are given by the  expression ðR ; N  ; P  Þ ¼

a12 a

K1 K2   þm þa13 P þm1 p 23 2

; a23 PK2þm2 ; P 

where P  is either zero

or a positive root of a third-degree polynomial. Proof. Critical points are given by the positive solutions to the system K1  R ða12 N  þ a13 P  þ m1 Þ ¼ 0;

ð2Þ

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K2  N  ða23 P  þ m2 Þ ¼ 0;

ð3Þ

P  ða31 R þ a32 N  þ m3 Þ ¼ 0:

ð4Þ

From (3), we have N ¼

K2 a23 P  þ m2

ð5Þ

and from (5) and (2) we obtain: R ¼

a12 a23 PK12þm2

K1 : þ a13 P  þ m1

Now from (4) we have two possibilities; either P  ¼ 0 or a31 R þ a32 N   m3 ¼ 0. Therefore, if a31 R þ a32 N   m3 ¼ 0 we obtain, after substitution a31

K1 K2 þ a32  m3 ¼ 0; a23 P  þ m2 a12 a23 PK2þm2 þ a13 P  þ m1

which is equivalent to require f ðP  Þ ¼ 0 where

m2 m1 m2 1 f ðP  Þ ¼ P 3 þ 2 ðm1 þ a12 N0 Þ þ ðk1  m3 ÞP 2 þ a23 a23 a23 a13

2 m2 m1 m2 1  ðm1 þ a12 N0 Þðk3  m3 Þ þ þ ðk2  m3 ÞP þ a23 a23 a23 a13 and k1 ¼

k2 ¼

a31 a13

K1 þ aa3223 K2 2 am232 þ am131

;

2 aa3113 K1 þ a32 N  1 a13



ðm1 þ a12 N  Þ þ

m2 a23



þ am231

m2 a23



þ am231

;

k3 ¼ a31 R þ a32 N  and the conclusion follows.

h

Note that there always exists an equilibrium point of the form ðR ; N  ; 0Þ. We will call this equilibrium the top predator-free equilibrium. The polynomial found in Theorem 1 can have up to three positive real roots that correspond to equilibrium points. The following results state the conditions under which these points are biologically feasible.

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Lemma 2. If m3 > maxfk1 ; k2 ; k3 g, then the open food web (1) will have only a unique non-trivial equilibrium point: the top predator-free state. Proof. Suppose m3 > maxfk1 ; k2 ; k3 g. Then we have that m3 > k1 , m3 > k2 , m3 > k3 ; therefore k1  m3 < 0, k2  m3 < 0, k3  m3 < 0. Under these conditions all coefficients are negative and by the Descartes rule of signs the polynomial does not have positive roots and the conclusion follows. h Lemma 3. If m3 < k3 , then there are at least two critical points. Proof. Suppose m3 < k3 , so k3  m3 > 0. Under this condition the coefficient of the third order term is negative and the independent term is positive. Applying Descartes rule we have at least one change of sign and therefore there is at least one positive root. h Lemma 4. If either m3 6 k2 or m3 P minfk1 ; k3 g then the number of critical points of the system is less than three. Proof. Assume m3 6 k2 or m3 P minfk1 ; k3 g. The polynomial f ðP Þ has roots of the form 0 1 K K 1 2 @ ; ; P  A; a12 a23 PK2þm2 þ a13 Pi þ m1 a23 Pi þ m2 i i

where f3 ðPi Þ ¼ 0, Pi a positive number and Pi 6¼ Pj if i 6¼ j. Note that f has the form f ðxÞ ¼ x3 þ bx2 þ cx þ d, were the coefficients b, c, d depend on the parameters k1 ,k2 ,k3 as defined in Theorem 1. Thus we can construct the system, bP12 þ cP1 þ d ¼ P13 ; bP22 þ cP2 þ d ¼ P23 ; bP32 þ cP3 þ d ¼ P33 or, in matrix notation, 0 2 10 1 0 3 1 b P1 P1 1 P1 @ P 2 P  1 A@ c A ¼ @ P 3 A: 2 2 2 d P32 P3 1 P33 Suppose now that 20 2 13 P1 P1 1 det 4@ P22 P2 1 A5 ¼ 0; P32 P3 1

ð6Þ

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so P12 ðP2  P3 Þ  P1 ðP2 þ P3 ÞðP2  P3 Þ þ P2 P3 ðP2  P3 Þ ¼ 0; ðP2  P3 Þ  ðP12  P1 ðP2 þ P3 Þ þ P2 P3 Þ ¼ 0; P12  P1 ðP2 þ P3 Þ þ P2 P3 ¼ 0; assuming all roots P1 , P2 , P3 , are different to each other. But then, the above implies

P1 ¼

P2 þ P3

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðP2 þ P3 Þ  4P2 P3 2

¼

P2 þ P3 ðP2  P3 Þ 2

and therefore P1 or P2 , a contradiction. Since the determinant cannot be zero then we must have 20 2 13 P1 P1 1 det 4@ P22 P2 1 A5 6¼ 0 P32 P3 1 and the unique solution to system (6) is given by: b ¼ P1 þ P2 þ P3 , c ¼ ðP1 P3 þ P1 P2 þ P2 P3 Þ, d ¼ P1 P2 P3 ; hence k2 6 m3 6 minfk1 ; k3 g, a contradiction and the result follows. h

3. Stability analysis In this section we proceed to study the local stability of the equilibrium points determined before. The jacobian matrix of the food web is 0 K1 1 a13 R  R a12 R A: J ðR ; N  ; P  Þ ¼ @ 0  NK2 a23 N  a31 P  a32 P  a31 R þ a32 N   m3 We have the following result: Theorem 5. If m3 > k3 then the top predator-free equilibrium point is locally asymptotically stable. Proof. The proof is an immediate consequence of evaluating the jacobian at this point

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0

 KR01 J ðR0 ; N0 ; P0 Þ ¼ @ 0 0 0 K1  R0 ¼@ 0 0 and the theorem is proved. Theorem 6. If

a13 K2 a23 a12 N 2

1 a13 R0 A a23 N 0 a31 R0 þ a32 N0  m3 1 a12 R0 a13 R0  KN02 a23 N0 A 0 k3  m 3 a12 R0  NK20 0

h

þ a31a32a12KR12 < 1 then the equilibrium point ðR ; N  ; P  Þ with

P  6¼ 0 is asymptotically stable. Proof. The jacobian matrix evaluated at a31 R þ a32 N   m3 ¼ 0, is 0 K1 1  R a12 R a13 R J ðR ; N  ; P  Þ ¼ @ 0  NK2 a23 N  A   a31 P a32 P 0

this

point,

assuming

with characteristic polynomial
 a13 K12 a32 K1 þ ; pðkÞ ¼ k  c2 k  c1 k þ P N R a31 a23 a12 1  a23 a12 N 2 a31 a12 R2 3

2







where ci > 0 is a combination of model parameters. Applying the Ruth– Hurwitz criteria we find that the critical point ðR ; N  ; P  Þ where P  6¼ 0 is asymptotically stable if a23aa1312KN2 2 þ a31a32a12KR12 < 1. h 3.1. An invariant infinite rectangle Up to this point we have shown the existence of multiple equilibrium points and the stability of each of them whenever they exist. However the question still remains as to the existence of at least one stable equilibrium point among the whole set of equilibria. To summarize we have the following results: 1. Whenever m3 > k3 the food web without the top predator is locally asymptotically stable. 2. Whenever m3 < k3 we have at least two critical points (one of them will be the point ðR ; N  ; 0Þ). 3. With the above assumption holding, the second equilibrium point will be locally asymptotically stable if a23aa1312KN2 2 þ a31a32a12KR12 < 1. In this section we proceed now to establish the boundedness of all solutions of our food web. We will do this by showing that all solutions to our system of

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283

equations remain in a certain region of R3 . In the following we will denote by x; y; z the solutions to or food- web model system (1). Let H ¼ fðx; y; zÞj 0 6 x 6 Km11 ; 0 6 y 6 Km22 ; z P 0g. The following theorem is straightforward and its proof will be omitted. Theorem 7. Let uðtÞ ¼ ðxðtÞ; yðtÞ; zðtÞÞ be a solution of our open food web model. If there exist t0 such that uðt0 Þ 2 H then uðtÞ 2 H for all t > t0 . Note that we have that 0 < R < Km11 , 0 < N  6 Km11 and P P 0 where ðR ; N  ; P  Þ is a critical point. We now proceed to characterize the solutions of our model system. Note that when z ¼ c ¼ const, we have that x0 > 0 whenever 

y<

K1 a13 c þ m1  a12 x a12

ð7Þ

and y 0 > 0 whenever y<

K2 ; a23c þ m2

ð8Þ

consequently, on any surface z ¼ c isoclines can be drawn on the xy plane by equating the right-hand sides of (7) and (8) to zero (see Fig. 1). The corresponding flux trajectories for z ¼ const are represented in Fig. 2. Now letting z take positive values we can obtain a curve such that for each z (Fig. 2), the curves x0 ¼ 0, and y 0 ¼ 0 intersect. This curve can be parametrized

Fig. 1. Intersection of the curves x0 ¼ 0 and y 0 ¼ 0 when letting z to take a constant-positive values.

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Fig. 2. Flux trajectories when z is constant, xðtÞ vs yðtÞ.

by only one parameter since both x and y coordinates can be expressed in terms of the third coordinate. Let this parameter be r. The curve is given by ! K1 K2 ; ;r : wðrÞ ¼ 2 a12 a Krþm þ a31 r þ m1 a23 r þ m2 32

3

We can now state the following lemma. Lemma 8. Let uðtÞ ¼ ðxðtÞ; yðtÞ; zðtÞÞ a solution to our open food web model. Then after certain time tp > 0, uðtÞ must remain in a neighborhood of the curve w  H. Now we will prove that all solutions to our food web model remain bounded for all time. Theorem 9. Let uðtÞ ¼ ðxðtÞ; yðtÞ; zðtÞÞ be a solution to our open food web model. If there exist t0 such that uðt0 Þ 2 H , then there exists M > 0 such that zðtÞ < M for all t > t0 . Proof. Suppose the opposite, that is, zðtÞ is riot a bounded function, so we have two cases: (1) limt!1 zðtÞ does not exists; (2) limt!1 zðtÞ ¼ 1. Suppose the first case is true. Then there exist a time sequence ftn g1 n¼1 such that limn!1 zðtn Þ ¼ limt!1 zðtÞ. From the equation for P we have z0 ¼ zða31 xðtÞ þ a32 yðtÞ  m3 Þ and thus

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R ln zðtÞ ¼ zðtn Þe

l0

ða31 xðsÞþa32 yðsÞm3 Þ ds

R ln

:

ða31 xðsÞþa32 yðsÞm3 Þ ds

So, limn!1 zðtn Þ ¼ limn!1 zðt0 Þe l0 zðtÞ is not bounded. So we have Z tn lim ða31 xðsÞ þ a32 yðsÞ  m3 Þ ds ¼ 1; n!1

285

¼ limt!1 zðtÞ ¼ 1 because

t0

then there exists M > 0 and N > 0 such that Z tn ða31 xðsÞ þ a32 yðsÞ  m3 Þ ds > M whenever n > N ; t0

but if we let f ðtÞ ¼ a31 xðtÞ þ a32 yðtÞ  m3 ; we have m3 < f ðtÞ < a31

K1 K2 þ a32 þ m3 m1 m2

and f could be positive or negative. Therefore, there exists a sequence fsn g1 n¼N þ1 such that tn1 < n 6 tn and Z tn Z sn f ðsÞ ds 6 f ðsÞ ds f ðsn Þ P 0: t0

t0

This implies that limn!1 zðsn Þ ¼ 1. Then, the points ðxðsn Þ; yðsn Þ; zðsn ÞÞ tend to be farthest to the curve wðrÞ when n is incremented but from lemma (1) we know that solutions remain near w for all r obtaining a contradiction. Now suppose the second case is true. A solution for the N equation satisfies y 0 þ yða32 zðtÞ þ m2 Þ ¼ K2 : Rl ða23 zðsÞþm2 Þ ds thus Let lðtÞ ¼ e l0 Rt K2 t0 lðsÞ ds þ yðt0 Þ : yðtÞ ¼ lðtÞ Rt Since limt!1 zðtÞ ¼ 1, then limt!1 yðtÞ ¼ 1, implying limt!1 t0 lðsÞ ds ¼ 1. Using L’Hopital rule it follows that Rt K2 t0 lðsÞ ds þ yðt0 Þ K2 lðtÞ ¼ lim lim yðtÞ ¼ lim t!1 t!1 t!1 ða23 zðtÞ þ m2 ÞlðtÞ lðtÞ K2 ¼ lim ¼ 0: t!1 a23 zðtÞ þ m2

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From the equation for R we have that its solutions must satisfy x0 þ xða12 yðtÞ þ a13 zðtÞ þ m1 Þ ¼ K1 ; letting Rl ða12 yðsÞþa13 zðsÞþm1 Þ ds nðtÞ ¼ e l0 ; we obtain xðtÞ ¼

K1

Rt t0

nðsÞ ds þ xðt0 Þ nðtÞ

and lim xðtÞ ¼ lim

t!1

K1

Rt t0

nðsÞ ds þ xðt0 Þ

¼ lim

nðtÞ K1 ¼ lim ¼ 0: t!1 a12 yðtÞ þ a13 ðtÞ þ m1 t!1

t!1

K1 nðtÞ ða12 yðtÞ þ a13 ðtÞ þ m1 ÞnðtÞ

Finally, from the equation for P we have Rl ða31 xðsÞþa32 yðsÞm3 Þ ds zðtÞ ¼ zðt0 Þe l0 ; but limt!1 a31 xðtÞ þ a32 yðtÞ ¼ 0; thus using the definition of limit and taking e ¼ m3 , it follows that there must exist a time Tm3 such that a31 xðtÞ þ a32 yðtÞ  m3 < 0 for all t > Tm3 . Therefore Z t Z Tm 3 ða31 xðsÞ þ a32 yðsÞ þ m3 Þ ds ¼ ða31 xðsÞ þ a32 yðsÞ þ m3 Þ ds t0

t0

þ Z

Z

t

ða31 xðsÞ þ a32 yðsÞ  m3 Þ ds

Tm3 Tm3

ða31 xðsÞ þ a32 yðsÞ þ m3 Þ ds t
0  K1 K2 6 a31 þ a32 m3 ðTm3  t0 Þ m1 m2

6

and thus  lim zðtÞ 6 zðt0 Þe

t!1

K



X

a31 m1 þa32 m2 m3 Tm3 ðTm3 t0 Þ 1

2

;

a contradiction and the theorem is proved.

h

Corollary 10. Let uðtÞ ¼ ðxðtÞ; yðtÞ; zðtÞÞ a solution of the open food web model, then

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lim xðtÞ 

K1 ¼ 0; a12 yðtÞ þ a13 zðtÞ þ m1

lim yðtÞ 

K2 ¼ 0: a23 zðtÞ þ m2

t!1

t!1

287

4. Simulations In this section we explore the dynamical behaviour of our food web system, particularly with respect to the existence and stability of multiple equilibrium points. Parameter values are shown in the table below and illustrate the case where there can be exactly four equilibria. K1 ¼ 10 K2 ¼ 0:1 a12 ¼ 0:002 a13 ¼ 0:00001 a23 ¼ 0:00001

a31 ¼ 0:00000005 a32 ¼ 0:00000005 m1 ¼ 0:0001 m2 ¼ 0:0001 m3 ¼ 0:000007200775

The four critical points are Ci ðR ; N  ; P  Þ; C0 ð49:97; 100; 0Þ; C1 ð58:28; 85:74; 1:66Þ; C2 ð85:88; 58:13; 7:20Þ; C3 ð7:20; 0:14; 6911:77Þ: We have two asymptotically stable points and two unstable points EQPT

Stability

Parameter relations

0

Unstable

1

Stable

2

Unstable

3

Stable

k3 ¼ 0:00149975 so m3 < k3 a13 K2 a32 K1 þ ¼ 1:47288 a23 a12 N 2 a31 a12 R2 a13 K2 a32 K1 þ ¼ 0:679375 a23 a12 N 2 a31 a12 R2 a13 K2 a32 K1 þ ¼ 239:796 a23 a12 N 2 a31 a12 R2

In this example we have multistability. Fig. 3 shows the asymptotically stable critical points for the variable P computed and simulated with Mathematica.

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Fig. 3. t vs ln P ðtÞ with different initial conditions; two estable critical points are shown corresponding to P1 ¼ 1:66 and P2 ¼ 6911:77.

5. Discussion We first comment briefly on food web models with more general functional responses and their resulting dynamics compared to the model analyzed in this paper. Throughout the paper, linear functional responses have been assumed to characterize predator–prey interactions at all levels in the food web. However, non-linearities in these interactions are probably the most common feature of natural food webs and previous theoretical work has shown the importance of such considerations on the dynamical properties of food webs [9,47]). Consequently, more realistic models of open food webs should incorporate the possibility of other types of functional responses. Here we comment on the behavior of open food webs with type II, saturating functional responses of predators to the availability of prey. 5.1. Type II functional responses Furthermore while recruitment of predators with open populations is generally independent of the local abundance of prey [7], the mortality rate of the predators settled into a given habitat must depend to some extent on the local availability of prey. To explore the effects of prey-dependent mortality rates on the dynamics of open food webs, it is possible to incorporate in our model mortality terms that increase total predator mortality when prey numbers are low, and asymptotically reach a maximum value equal to the density independent mortality rate, m, when prey are abundant.

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A completely open food web model which includes both type II functional responses and prey-dependent mortality rates would be dR a12 RN a13 RP ¼ K1    m1 R; dt 1 þ a12 p12 R 1 þ a13 p13 R þ a23 p23 N   dN a23 NP d2 ¼ K2   m2 N 1 þ ; dt 1 þ a13 p13 R þ a23 p23 N 1 þ a12 ~p12 R   dP d3 ¼ K3  m3 P 1 þ ; dt 1 þ a13 ~ p13 R þ a23 ~ p23 N where pij are coefficients expressing the contribution of an average individual of prey i to the saturation (satiation) of predator j, di , is the increment in mortality rate produced by the loss of predation opportunities, and ~pij express the contribution of prey species i to reducing mortality rate of predator j. Notice that the threshold for consumption is different than the threshold for mortality in this model. This is conceivably possible if the basic metabolic rate of the species, i.e. the amount of calories per unit time necessary to maintain basic body functions and simply stay alive, is different from (and less than) the amount of calories required to satisfy all of the animals needs (e.g. movement, reproduction), which would completely be achieved at satiation levels. Therefore, one would expect the threshold for mortality to be lower than that for consumption. Given the large number of parameters, we limit ourselves here to explore two limiting cases: the case where there is no threshold for consumption (type I functional response), or it is so much greater than the threshold for mortality as to be irrelevant, given by p13 ¼ p23 ¼ p12 ¼ 0; di > 0 and the case where the threshold for consumption and the threshold for mortality are the same, given by p13 ¼ p~13 > 0; p23 ¼ p~23 > 0; p12 ¼ p~12 > 0; di > 0: Mortality threshold effects: The completely open model with linear functional responses and prey-dependent mortality thresholds is dR ¼ K1  a12 RN  a13 RP  m1 R; dt

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  dN d2 ¼ K2  a23 NP  m2 N 1 þ ; dt 1 þ a12 p12 R   dP d3 ¼ K3  m3 1 þ : dt 1 þ a13 p13 R þ a23 p23 N Analysis of this model is limited to computer simulation as analytical expressions for the equilibrium point(s) are difficult to obtain and not particularly informative. The addition of the saturation terms does not greatly affect the overall behavior of the model. There exist a single three-species equilibrium point and under a wide range of parameter choices, all initial conditions flow to this single point attractor. Functional responses and prey-dependent mortality: As mentioned above, we next looked at the behavior of a food web in which the predators have type II functional responses, prey-dependent mortality rates, and the thresholds for consumption and mortality are the same. The model is dR a12 RN a13 RP ¼ K1    m1 R; dt 1 þ a12 p12 R 1 þ a13 p13 R þ a23 p23 N   dN a23 NP d2 ¼ K2   m2 N 1 þ ; dt 1 þ a13 p13 R þ a23 p23 N 1 þ a12 p12 R   dP d3 ¼ K3  m3 P 1 þ : dt 1 þ a13 p13 R þ a23 p23 N Despite the existence of non-linear terms, our computer simulations (not shown) indicate that this model also had a single, stable three-species equilibrium point. The addition of saturating functional responses to the predation terms does not cause the limit cycle and chaotic behavior typically observed in food webs with closed populations [9,15]. This result highlights the strong stabilizing effects of having open population structure at local scales. 5.2. Linear functional response We come back now to the main conclusions derived from the analysis of the model with linear functional response. The system with basal and intermediate predator species open and the top predator closed, can have a maximum of four critical points and at least one corresponding to the omnivore-free equilibrium point. In this case our system reduces to a two level food web without omnivory. This equilibrium point has stability properties that depend on the capacity of invasion of the omnivore species. If the omnivore succeeds in invading the community then a three level food web can be established but with more complex stability properties. When the two species food web is unstable, that is when the equilibrium point without the top predator is unstable, then

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there may exist three more critical points at least one of which is asymptotically stable. The exact number of critical points that may exist depend on the food web parameter space. The question of bistability is still open. The parameters k1 , k2 and k3 play a crucial role in determining the number of equilibrium points that our food web system has. The parameter k1 is a weighted ratio of total recruitment rate to total mortality rate in the food web levels that present open recruitment: Unfortunately k2 is a very difficult parameter to interpret. It relates the recruitment rate of the basal species with the equilibrium value of the intermediate species, both of them with open recruitment. Of course there is a dependence on the predation rates but of a complex nature. Finally k3 is the weighted average of the equilibrium densities of the basal and the intermediate species. All three parameters determine the existence and number of biologically feasible equilibrium points and involve the top predator only through predation rates, its biomass conversion efficiency into predator numbers but not through its density at equilibrium. Our numerical simulations show the existence of two asymptotically stable feasible equilibrium points in sharp contrast with the classical local recruitment Lotka–Volterra food web models analyzed by Holt and Polis 11] or McCann et al. 15]. Our results, as far as local dynamics and numerical simulations can show, indicate that there can be no chaotic dynamics in this particular type of open food web. Dependence on initial conditions is however crucial in this case. In our numerical results one of the three-species equilibrium points is characterized by relatively low densities in all three species, whereas the other possessed densities much higher, which could imply higher robustness against stochastic perturbations. Overall, our results show that three species stability in these food webs is critically constrained by a number of attributes of the interacting species and rates of recruitment of open populations. We speculate that these constraints could explain the scarcity in natural marine communities of top predators with closed populations feeding on other predators and prey with open-recruitment. The system conformed by the seastars Pisaster ochraceus (with long lived planktonic larvae) and Leptasterias hexactis (direct developer), which feed on mussels and barnacles could represent such an example in the northeastern Pacific, but there is no predation by Leptasterias on Pisaster [16]. The subtidal soft bottom communities of southern Chile dominated by the large predatory gastropod Chorus giganteus, which has limited dispersal and preys on species with free swimming larvae ([49]), could be a model system to evaluate the predictions of our model. Our results also suggest that local food webs conformed by species with open populations lack complex dynamics and are essentially stable but with a dependency on initial conditions unseen on more classical food webs. Inclusion of a trophic level with local reproduction (the omnivore predator in our system) constrains conditions for stability of the three-species equilibrium, as compared to a completely open food web, but it does not produce complex

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behaviors. We believe the models confidently show that local biological interactions are not sufficient to produce complex dynamics, as opposed to closed systems. However, our model makes an unwarranted assumption: recruitment among the open species is a constant and there is no explicit spatial dependence included. The models predict the conditions for stable coexistence, which can be qualitatively evaluated against field information. In the case of the specific life history combinations modeled here, it appears that there are few opportunities for stable coexistence in natural systems. Most studies of recruitment have found extremely variable recruitment in benthic communities (e.g. [2,18,21,48], suggesting that the assumption of constant recruitment is violated more often than not. Also, many of the communities that present recruitment uncoupled from local reproduction have also a very strong spatial structure which may unstabilize the system. Indeed one of the interesting aspects of our models is that they indicate that unpredictable or chaotic dynamics in open communities can only be due to variable recruitment or spatial structure. Future theoretical research into the dynamics of local open systems should thus explore the consequences of variable recruitment and of different combinations of dispersal abilities on the overall stability of the system. Acknowledgements Ismael Velazquez acknowledges support of a doctoral CONA CYT grant; JXVH acknowledges support from grant IMP D. 00154. References [1] C.S. Briscoe, K.P. Sebens, Omnivory in strongylocentrotus droe-baliiensis (M€ uller) (Echinodermata: Echinoidea): predation on subtidal mussels, J. Exp. Mar. Biol. Ecol. 115 (1988) 1–24. [2] M.J. Caley, M.H. Carr, M.A. Hixon, T.P. Hughes, G.P. Jones, B.A. Menge, Recruitment and the local dynamics of open marine populations, Ann. Rev. Ecol. Syst. 27 (1996) 477–500. [3] S. Connolly, J. Roughgarden, Theory of marine communities: competition, predation and recruitment-dependent interaction strength, Ecol. Monogr. 79 (1999) 277–296. [4] S. Diehl, Fish predation and benthic community structure: the role of omnivory and habitat complexity, Ecology 73 (1992) 1646–1661. [5] W.F. Fagan, Omnivory as a stabilizing feature of natural communities, Am. Nat. 150 (1997) 554–567. [6] S.D. Fretwell, Food chain dynamics: the central theory of ecology, Oikos 50 (1987) 291–301. [7] S.D. Gaines, K.D. Lafferty, Modeling the dynamics of marine species: the importance of incorporating larval dispersal, in: L. McEdward (Ed.), Ecology of Marine Invertebrate Larvae, CRC Press, New York, 1995, pp. 389–412. [8] N.G. Hairston, F.E. Smith, L.B. Slobodkin, Community structure, population control, and competition, Am. Nat. 94 (1960) 421–425. [9] A. Hastings, T. Powell, Chaos in a three species food chain, Ecology 72 (1991) 896–903. [10] R.D. Holt, Predation, apparent competition, and the structure of prey communities, Theor. Pop. Biol. 12 (1977) 197–229.

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