Periodic Traveling Waves Generated by Invasion in Cyclic Predator--Prey Systems: The Effect of Unequal Dispersal

Abstract

Periodic traveling waves (wavetrains) have been an invaluable tool in the understanding of spatiotemporal oscillations observed in ecological data sets. Various mechanisms are known to trigger this behavior, but here we focus on invasion, resulting in a predator-prey-type interaction. Previous work has focused on the normal form reduction of PDE models to the well-understood λ-ω equations near a Hopf bifurcation, though this is valid only when assuming an equal rate of dispersion for both predators and prey-an unrealistic assumption for many ecosystems. By relaxing this constraint, we obtain the complex Ginzburg-Landau normal form equation, which has a one-parameter family of periodic traveling wave solutions, parametrized by the amplitude. We derive a formula for the wave amplitude selected by invasion before investigating the stability of the solutions. This gives us a complete description of small-amplitude periodic traveling waves in the governing model ecosystem.

A 4-Dimensional Mathematical Model for Interaction between the Human Immune System and a Virus

“…In this work we aim to propose a mathematical model to study the interaction between a virus and the human immune system represented by following three components: the innate immunity, the humoral immunity (after passing by infection or by vaccination) and the cellular immunity (only after passing by infection). Our approach is based on a modified predator-prey methodology ( [7], [8], [9], [10], [11], [12]) used in population dynamics. It is need to change some initial hypotheses used in classical predator-prey models to take into account the types of interactions between the immune system and the virus (predator).…”

Heterogeneity in Behaviour and Movement can Influence the Stability of Predator–Prey Periodic Travelling Waves

“…In biological systems, predator diffusion is typically larger than prey diffusion, so the constraint may appear overly constrictive. Indeed, the ratio between prey and predator diffusion is known to have an effect on the stability boundary of the PTWs in homogeneous systems (Smith and Sherratt 2007 ; Bennett and Sherratt 2017 ). However, can be satisfied while still allowing predator diffusion to be larger than prey diffusion at the patch level.…”

“…A natural question is whether our conclusions involving the effects of heterogeneity in movement and behaviour on PTW stability continue to hold for larger-amplitude PTWs. Nonetheless, the investigation of small amplitude is still used in the literature (Bennett and Sherratt 2017 , 2019 ) as it provides useful analytical insights and limiting case predictions. Moreover, it produces a valuable framework to numerically investigate larger amplitude scenarios and a guideline to explore the parameter space through other non-analytical methods (e.g.…”

“…Spatiotemporal data is often challenging to obtain (Bennett and Sherratt 2017 ), since it involves a laborious and costly monitoring at several locations through usually extensive periods of time (Berthier et al. 2014 ).…”

A Local Analysis of a Mathematical Pattern for Interactions between the Human Immune System and a Pathogenic Agent

“…The main aim of this paper is to construct a four-dimensional deterministic pattern to study the interaction between a virus and the human immune system, and the pattern is represented by the following three components: innate immunity, humoral immunity (after passing the infection or by vaccination), and cellular immunity (only after passing infection). This approach is new, although it is based on a modified predator-prey methodology used in population dynamics [8][9][10][11][12][13]. Some initial hypotheses used in classical prey-predator patterns need to be changed in order to take into account the types of interactions between the immune system and the virus (predator).…”