Abstract
The Rosenzweig-MacArthur (1963) criterion is a graphical criterion that has been widely used for elucidating the local stability properties of the Gause (1934) type predator-prey systems. It has not been stated whether a similar criterion holds for models with explicit resource dynamics (Kooi et al. (1998)), like the chemostat model. In this paper we use the implicit function theorem and implicit derivatives for proving that a similar graphical criterion holds under chemostat conditions, too.
1. Introduction
Several ecological phenomena are studied under chemostat conditions; compare Smith and Waltman [1]. A phenomenological model containing such a situation is given by Here is the substrate, is the prey having the substrate as its limiting resource, and is a predator feeding on the prey . The parameters , , , , , , , and stand for concentration, dilution rate, search rate for the prey, handling time for the prey (cf. [2]), conversion factor for the prey, search rate for the predator, handling time for the predator, and conversion factor for the predator, respectively.
A reduction into two dimensions [3] is often made when studying (1) and related systems. More precisely, consider the functionIt satisfies meaning that the surface is asymptotically invariant for (1). A study of the system on this surface allows for reducing it to a planar predator-prey system as follows:Such reductions can be made rigorously under certain conditions; see Smith and Waltman [1]. As an example for how such a procedure may break down Thieme [4] gave the following example in cylindrical coordinates , , , , :with initial data , , and positive constant parameters. In this case, consider the functionAlso here, we havemeaning that the surface is asymptotically invariant for (4) and a study of (4) on this surface should allow for reducing it to the planar systemIn Cartesian coordinates, (4) has three equilibria , , that are illustrated by -marks in Figure 1. All solutions are attracted towards the unit circle. Solutions with initial conditions in the plane has some equilibrium on the unit circle as its limit set. But if , then the limit set is the whole unit circle. We see, however, that the chain recurrent set [5] is the whole unit circle (and the origin) and does not depend on initial conditions.
Back to (3), if , then the growth function of (3) is given byin the absence of predators. Therefore, growth function is unimodal on the intervalprovidedThe last inequality is identically true. We shall later use the results above to introduce relevant coordinate-transformations for (3).
Equivalents of the predator-prey system (3) have been studied in Smith and Waltman [1] and Kuang [6]. The results were that local stability implies global stability and that uniqueness of limit cycles was proved for a certain range of parameters. It is still not known whether the limit cycle is unique for all parameters of (3) and a further analysis and improvement of these results remain outside the scope of this paper.
2. A Related Gause [7] Type Predator-Prey System
We start this study by relating (3) to a widely used class of predator-prey systems. Assuming that , we getand this model can be identified as a Gause [7] type predator-prey model on the isocline formwithsee Lindström and Cheng [8]. In general, the conditions on the involved functions are stated as (A-I), , and are ,(A-II), for ,(A-III) for ,(A-IV), ,
and it is easy to see that functions (13) meet criteria (A-I)–(A-IV) and that the solution of (the predator isocline) in this case gives If (A-I)–(A-IV) and , then solutions of system (12) remain positive and bounded [8]. Moreover, it has three equilibria: which is a saddle, also a saddle, and finally that has the JacobianWe have and with , so the Trace-determinant criterion [9] gives immediately the classical Rosenzweig-MacArthur [10] criterion stating that the interior equilibrium is locally asymptotically stable when the predator isocline intersects the prey isocline at point where the prey isocline decreases and is unstable when . In fact, all the topological properties including results of global stability and uniqueness of limit cycles for all parameters of (11) are known; see Lindström and Cheng [8].
3. Reparametrization
We reparameterize the system in order to eliminate some of the parameters involved (see, e.g., [11]). We remember the growth interval (9) and introduce the new variables, and the new parameters,Now, (3) takes the formwith , , and ( ensures a two-species food-chain) and finally the chemostat estimateThe case corresponds to the known case (see Lindström and Cheng [8]) and the main purpose of this paper is to derive a Rosenzweig-MacArthur [10] criterion for (18) when . For the variables we assume , .
We notice that the alternative transformations , , , , , , give system (1.2) in Kuang [6]. Therefore, the system under study is the same but the set of feasible parameters might be differently identified. Our reparametrization is more complicated. However, the properties of (3) suggest transformation (16) since formula (16) provides a normalization of the growth interval for in (9) and the intervalfor into the unit interval for both variables and .
4. Isocline Form and Properties of Equilibria
We rewrite the system on a form allowing isoclines to be analyzable. It is far from clear how this should be done in the chemostat case. We decided to work with the following form:and state our conditions on the involved functions as (C-I), , , and are ,(C-II), for ,(C-III), ,(C-IV), ,(C-V), , ,(C-VI), .
We note that system (18) corresponds to the choiceand that this choice meets conditions (C-I)–(C-VI). In particular, we have (C-III) sinceand (C-IV) sinceand the last inequality holds simply because . Before going further, we prove a basic theorem.
Theorem 1. Consider the bounded set , , . Solutions of (21) starting in this set remain there.
Proof. By uniqueness of solutions no solutions can intersect the four solutions , , , , , and . Thus, solutions remain positive. To prove that solutions remain bounded, we assume and consider the series of inequalities
We further conclude thatso we work with decreasing and concave down as far as possible, too. We have one equilibrium at the origin, one at the carrying capacity , and one equilibrium at , where satisfies the condition . We prove first, that the first two equilibria are saddles. The corresponding Jacobians are given byWe note that also contains the information that the eigenvector corresponding to the positive eigenvalue points into the triangle , , and . This criterion can be formulated aswhich is true due to and .
5. Implicit Functions and Our Criterion
For the interior equilibrium we start by doing some estimates concerning its location and define an implicit function for the prey isocline by the equationBy the implicit function theorem, we getThe denominator of the above expression is always positive and therefore the implicit function is defined for . The sign of the derivative of the implicit function is defined by the nominator. We start computing special values of this implicit function and conclude that andWe now go on computing the Jacobian of (21) at the interior equilibrium and get We have thatso the eigenvalues have the same sign and their stability is determined by the trace only. This is entirely in concordance with index theory [12, 13] asserting that the index of all fixed points in the interior is 1. We haveand conclude that the sign of this expression agrees with the sign of (31). Thus, we have a Rosenzweig-MacArthur [10] criterion for the chemostat. We summarize our conclusions in the following theorem.
Theorem 2 (a Rosenzweig-MacArthur [10] graphical criterion for the chemostat). Assume (C-I)–(C-VI). The interior fixed point of (21) is locally stable when and unstable when . When , the chemostat system has at least one limit cycle. The prey isocline decreases at the vicinity of 1 and is located in the bounded set , , for all .
The last assertion is due to the Poincaré-Bendixson theorem (see, e.g., [9]) because the triangular set , , is invariant and its boundary is not approached by any of the solutions. We illustrate the graphical conclusions of Theorem 2 in Figure 2.
(a)
(b)
Finally, we return to the specific expressions that we used as our prototype example (18) in order to check what additional conclusions can be made. We note thathas one solution in the unit interval and one solution greater than 1 (insert and , resp., in the above equation and remember the chemostat estimate (19)). We are interested in the solution in the unit interval and it turns out to beWe also conclude thatbecause of (19). We continue with an analysis of the shape of this curve. The defining equation (30) can be written as which after some refinement is the level curve of a quadratic form given byWe note immediately that the quadratic form at the right hand side must be indefinite (both leading terms and are of different sign). The level curve consists therefore of either two intersecting lines or a hyperbola. One branch of this hyperbola is the predator isocline curve and when the involved functions are given as in (22), the isocline curve is given explicitly byThe minus sign was selected in front of the square root above since we require . We conclude that this isocline is concave down and .
Theorem 3. The prey isocline for our chemostat system (21) with (22) is a concave down function corresponding to a branch of a hyperbola.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
The authors thank Sebastian Thörngren for checking a substantial part of the results in a preliminary version of this paper in his Bachelor’s thesis [14].