On the asymptotic behaviour of solutions of an asymptotically Lotka – Volterra model

We make more realistic our model [Nonlinear Anal. 73(2010), 650–659] on the coexistence of fishes and plants in Lake Tanganyika. The new model is an asymptotically autonomous system whose limiting equation is a Lotka–Volterra system. We give conditions for the phenomenon that the trajectory of any solution of the original nonautonomous system “rolls up” onto a cycle of the limiting Lotka–Volterra equation as t → ∞, which means that the limit set of the solution of the non-autonomous system coincides with the cycle. A counterexample is constructed showing that the key integral condition on the coefficient function in the original non-autonomous model cannot be dropped. Computer simulations illustrate the results.


Introduction
A group of scale-eating cichlid fishes from Lake Tanganyika provide an interesting and well known example of an evolved asymmetry in vertebrates [5].Members of the Perissodini tribe of these fishes have evolved dental and craniofacial asymmetries as a result of which they obtained an increased efficacy to remove scales from the left or right flanks of prey.This means that one morphological group of the fishes have their mouth parts twisted to the left, thus they can better eat scales off their prey's right flank.The other morph, whose mouth is turned to the right, eats scales off its prey's left flank.
In [1], the authors investigated a non-autonomous model which describes the change in time of the amount of two fish species -a herbivore and a carnivore -living in Lake Tanganyika and the amount of the plants eaten by the herbivores.The model considers n groups of the carnivore fish, corresponding to n different morphs.The model consists of two parts: reproduction taking place at the end of each year is described by a discrete dynamical system, while the development of the population during a year is described by a non-autonomous system of differential equations.The authors assumed that the whole system of the nutrition chain consisting of plants, herbivores and carnivores is supported by the constant energy flow provided by the Sun.
In several steps, the original system was transformed into the following equation: where L(t) corresponds to the amount of plants and G(t) corresponds to the prey fish.The function λ(t) is monotonically decreasing and tends to a positive constant λ * exponentially as t → ∞.In [1], it was shown that the point (λ * , c/λ * ) is an eventually uniform-asymptotically stable point in the large of (1.1) on the quadrant {(L, G) : L ≥ 0, G > 0}.
In the present paper we make the above model more realistic: instead of assuming a constant energy flow, we consider exponential growth for the plants in the absence of the herbivores.Under this condition the first equation of (1.1) changes and we obtain the new model This is an asymptotically autonomous system, which has a special case of the classical Lotka-Volterra predator-prey model as the limiting equation.In this paper we show that the limit set of all solutions of (1.2) is a solution of the Lotka-Volterra-type limiting equation.

Numerical experiments
For a numerical simulation we first consider equation (1.2) with λ(t) chosen as (for details, see [1, Section 2]) It is easy to see that the limiting equation in this case takes the form As shown in Figure 2.1, numerical simulation of solutions of the two systems (1.2) and (2.1) suggests that trajectories of (1. 2) "roll up" on cycles of the limit system (2.1).

The results
Consider the classical Lyapunov function [2] V(L, G) to the Lotka-Volterra equation It is well-known [2] that the level curves are Jordan curves around the equilibrium (λ * , c), which are the trajectories of the solutions.
In other words, all solutions are periodic; i.e., all trajectories are cycles.Let a piecewise continuous function λ : R Then equation (3.2) is the limiting equation of the original equation The derivative of (3.1) with respect to the original equation (3.5) is Then for every solution t → (L(t), G(t)) of (3.5) the finite limit Proof.By (3.6) there exists a K such that V(L, G, t) ≤ K|λ(t) − λ * | holds for all (L, G) ∈ (R + ) 2 and t ∈ R + .Therefore, for every pair t 1 < t 2 we have Using condition (3.7) we obtain that for the function v(t) := V(L(t), G(t)) the following condition is satisfied: for every ε > 0 there exists a T such that t 1 , By the Cauchy criterion on the convergence the limit lim t→∞ v(t) = V * exists.
To prove our main theorem we recall some definitions and results from stability theory.Consider the general system of differential equations ẋ = f (t, x) (3.9) with f : R + × Ω → R n , where Ω is an open subset of R n ; 0 ∈ Ω.Let • denote any norm in R n .Suppose that for every t 0 ≥ 0 and x 0 ∈ Ω there exists a unique solution x(t) = x(t; t 0 , x 0 ) of equation (3.9) for t ≥ t 0 satisfying the initial condition x(t 0 ; t 0 , x 0 ) = x 0 .
A point x * ∈ Ω is said to be a positive limit point of a solution x of (3.9) if there exists a sequence t j such that t j → ∞ and x(t j ) → x * as j → ∞.The set of all positive limit points of x is called the positive limit set of x and is denoted by Λ + (x).
The translate of a function f : R + × Ω → R n by a > 0 is defined as f a (t, x) := f (t + a, x).The function f is called asymptotically autonomous if there exists a function f * : Ω → R n such that f a (t, x) → f * (x) as a → ∞ uniformly on every compact subset of R + × Ω. f * and ẋ = f * (x) will be called a limit function and a limiting equation, respectively.
Let f (t, x) be asymptotically autonomous.A set F ⊂ Ω is said to be semi-invariant with respect to equation (3.9) if for every (t 0 , x 0 ) ∈ R + × F there is at least one non-continuable solution x * : (α, ω) → R n of the limiting equation ẋ = f * (x) with x * (t 0 ) = x 0 such that x * (t) ∈ F for all t ∈ (α, ω).

Theorem A ([4]
). Suppose that f is asymptotically autonomous.Then for every solution x of equation (3.9) the limit set Λ + (x) ∩ Ω is semi-invariant.L. Markus [3] proved (see also [6]) the following generalization of the Poincaré-Bendixson theorem: if n = 2, then the positive limit set of a forward bounded solution of an asymptotically autonomous equation either contains equilibria of the limiting equation or is the union of cycles of the limiting equation.Our main result gives an analogous, more precise theorem for (3.5) without assuming forward boundedness.Theorem 3.3.Suppose that the conditions Then for every solution t → (L(t), G(t)) of the original non-autonomous equation (3.5) the solution tends to (λ * , c) as t → ∞, or there is a cycle γ of the Lotka-Volterra limiting equation (3.2) such that the curve t → (L(t), G(t)) "rolls up" onto the γ as t → ∞, which means that It is easy to see that Λ + (L, G) ⊂ γ.On the other hand, by Theorem A, Λ + (L, G) is semiinvariant with respect to (3.5).Now this means that γ ⊂ Λ + (L, G), so γ = Λ + (L, G).
The following question arises: is every trajectory of the Lotka-Volterra limiting equation (3.2) (included the equilibrium (λ * , c)) the limit set of some solution of the original equation (3.5)?Conjecture 3.4.Every cycle of the Lotka-Volterra limiting equation (3.2) is the limit set of some solution of the original equation (3.5).

A counterexample
In this section we construct an example showing that condition (iii) in Theorem 3.3 is essential in the sense that without (iii) the theorem is not true.The suitable coefficient λ(t) will be a step function.The function will be constructed dynamically step by step: where the sequences 0 =: have to be found so that equation (3.5) with (4.1) have an unbounded solution.The kth piece of the trajectory of the desired unbounded solution will be a "half" of a cycle of the Lotka-Volterra equation with λ = λ k (see Figure 4.1).At first we will fix λ * and {λ k } ∞ k=1 properly, and require the initial condition G(t 0 ) = c.Then, by the method of the mathematical induction, if t 1 , . . ., t k−1 are already defined, then we choose t k so that G(t k ) = c be satisfied, i.e., so that the piece of the trajectory over the interval [t k−1 , t k ] be a half of a cycle of (4.2) with λ = λ k .

Figure 2 . 1 :
Figure 2.1: Trajectories of the original system (blue dashed line) and the limiting equation (thin red line)

Figure 4 . 1 :
Figure 4.1: The result of the first two steps of the procedure

Figure 4 . 3 :
Figure 4.3: The trajectory of the unbounded solution