Simple dynamics in non-monotone Kolmogorov systems

Lei Niu,Alfonso Ruiz-Herrera

doi:10.1017/prm.2021.75

Abstract

In this paper we analyse the global dynamical behaviour of some classical models in the plane. Informally speaking we prove that the folkloric criteria based on the relative positions of the nullclines for Lotka–Volterra systems are also valid in a wide class of discrete systems. The method of proof consists of dividing the plane into suitable positively invariant regions and applying the theory of translation arcs in a subtle manner. Our approach allows us to extend several results of the theory of monotone systems to nonmonotone systems. Applications in models with weak Allee effect, population models for pioneer-climax species, and predator–prey systems are given.

Highlights

The discrete equation xn+1 = xng(xn), n ∈ Z+={0, 1, 2, . . .}, (1.1)is a popular modeling framework for analysing the dynamical behaviour of a single species
We prove that the relative position of L1 and L2 completely determines the dynamical behaviour of (3.1)
It is well known that the relative position of the nullclines determines the dynamical behaviour of the classical Lotka–Volterra model x = x(r1 − x − αy) y = y(r2 − y − βx)

Summary

Introduction

Is a popular modeling framework for analysing the dynamical behaviour of a single species. In (1.1), xn 0 is the population density in the n-th generation and g(xn) 0 represents the density-dependent growth rate (or fitness function) from generation to generation. A common assumption in population dynamics is that g is decreasing. This means that the growth rate is mainly determined by negative density-dependent mechanisms such as intra-specific competition [5] or cannibalism. In this paper we describe the global dynamical picture of model (1.2) when the functions gi are not necessarily decreasing. Our criteria could be perceived as an extension of Hou’s results to non-monotone systems. To show the versatility of our results with different interactions, we discuss the dynamical behaviour of several classical predator–prey systems. We conclude the paper with a discussion on our findings

Mathematical framework

Discussion