# A survey on stably dissipative Lotka-Volterra systems with an application to infinite dimensional Volterra equations

Abstract: For stably dissipative Lotka{Volterra equations the dynamics on the attractor are Hamiltonian and we argue that complex dynamics can occur. We also present examples and properties of some infinite dimensional Volterra systems with applications related with stably dissipative Lotka-Volterra equations. We finish by mentioning recent contributions on the subject.

## Summary (2 min read)

### 1. Introduction

- He investigated in considerably detail the association of two species, one of which (the predator) feeds on the other (the prey) and for this case the authors have the nowadays called Lotka–Volterra equations (see [29, p. 14, eq(4)]).
- The dynamics of systems of type (1) are far from understood, although special classes of these Lotka–Volterra systems have been studied.
- While seeking a variational principle for the system, he was successful in finding a Hamiltonian formulation in the case where the interaction matrix is skew-symmetric, at the expense of doubling the number of dimensions (see Section 2 for details).
- It was already observed in [23] that there may exist periodic orbits on (non-trivial) attractors.

### 2. Basic notions

- Here, the authors will recall some basic notions and facts concerning general Lotka–Volterra systems which will be useful in the next sections.
- For a more detailed account of general properties of Lotka–Volterra systems the authors refer to the book by Hofbauer and Sigmund [9].
- Similarly, to exclude α-limit points one uses the Liapunov function −V .
- On the other hand, the following result shows that the average behavior of the orbits is related to the values of the fixed points (see [2]).

### 3. Conservative systems

- In the case were system (1) is conservative Volterra was able to introduce a Hamiltonian structure for the system by doubling the number of variables.
- The authors recall now Volterra’s construction, so they assume that system (1) is conservative and a choice of gauge has been made so that the matrix (ajk) is skew-symmetric.
- The full justification of this procedure will be given later in the section.
- Now, if one introduces another set of variables Recall that the modern approach to Hamiltonian systems is based on the following generalization of the notion of a Poisson bracket (see for example [1], [17]).

### 4. Dissipative systems

- Since the authors want their results to persist under small perturbation they introduce the following definition.
- Note that the authors only allow perturbations that have the same graph as the original system.
- Also they use instead the name stably admissible.
- For stably dissipative systems this choice can be improved [24]: Lemma 4.2.
- It will be convenient to modify slightly the notion of graph associated with the system the authors introduced above.

### 5. Examples

- Example 5.1. In [3, p. 159, eq. (33)], the authors give a detailed analysis of a 4-dimensional Lotka–Volterra system in order to illustrate the complexity of the dynamics that can occur on the attractor.
- These equations were extensively analysed by Duarte, Fernandes and Oliva in [3] where the flow generated by (51) was discussed using a reduction procedure to establish the existence of invariant sets with a Hamiltonian structure.
- Theorem 5. Consider the system of Lotka–Volterra equations (61) coupled with the linear equations (62) and assume that: (i) the Lotka– Volterra system (61) has a singular point q ∈ Rm+ ; and, (ii) the interaction matrix A is stably dissipative.
- The paper [31] by Zhao and Luo proves necessary and sufficient conditions for a matrix to be stably dissipative.

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