1. Existence of Wavetrains with Periods
Numerical bifurcation analysis of wavetrains in a RD type two species predator-prey model
Md. Muztaba Ahbab*, Md. Ariful Islam Arif, Afia Farzana, Popy Das, Taslima Khatun and M. Osman Gani
Department of Mathematics, Jahangirnagar University, Savar, Dhaka, Bangladesh *
Abstract
Existence of Wavetrains with Periods
We consider a two-variable reaction-diffusion system of equations model to understand the mechanism of irregular behavior of predator and prey. We study the numerical existence and stability of the periodic traveling waves (wavetrains) in the model. A key feature of our study is subdivided the parameter plane into the stable and unstable region through a stability boundary. We calculate essential spectra of the wavetrains to understand the stability of the waves. The irregular behavior of predator and prey is found to occur when the solutions cross the stability boundary.
Introduction
Population models, take a place in several fields of mathematical biology, have been proposed and studied broadly throughout the world because of the global existence and importance in mathematical and biological point of view [1]. One of the prevalent affairs in ecology is the progressive connection between predator and its prey. Significant histories have been carrying through the field of theoretical ecology by the reaction-diffusion models. An idiosyncratic trend of the predator-prey system is cyclic [2]. The Holling-Tanner model is the most extensively used population model which plays an eminent role in the prospect of the interesting dynamics it brings up [3]. In order to study a reaction-diffusion equation for population dynamics, we consider the Holling-Tanner model [4] for a predator-prey system, regarding for diffusion of both species as well as nonlocal prey intraspecies competition which is in between individuals of the same species. We use the method of continuation to study these solutions in one dimension via the continuation package WAVETRAIN [5]. We investigate the existence of periodic traveling waves (wavetrains) for the proposed model with stability. This enables us to understand the behavior of wavetrains in one dimension.
Mathematical Model
(1)
(2)
Model-Aided Understanding
References
u: the population of prey, v: the population of predator species.
We consider the parameter as a bifurcation parameter.
Steady State:
[1]Berryman, Alan A. "The Orgins and Evolution of Predator‐Prey Theory." Ecology 73.5 (1992):
[2]Sherratt, J. A. "Periodic travelling waves in cyclic predator–prey systems." Ecology Letters 4.1 (2001):
[3]Zhang, Jia-Fang. "Bifurcation analysis of a modified Holling–Tanner predator–prey model with time delay." Applied Mathematical Modeling 36.3 (2012):
[4]Tanner, James T. "The stability and the intrinsic growth rates of prey and predator populations." Ecology 56.4 (1975):
[5]. J.A. Sherratt, “Numerical continuation methods for studying periodic travelling wave (wavetrain) solutions of partial differential equations, Applied Mathematics & Computation 218, (2012).
No PTWs
L=100.7
delta=0.7
c=5
L=403.1 delta=0.7c=20
Existence of PTWs
Bifurcation Diagrams
Stability of Wavetrains (PTWs)
a stable PTW
an unstable PTW
Stability boundary
Hopf locus
Concluding Remarks
The behavior of PTW solutions studied here for our consider model (1). We have numerically calculated wave existence, wave stability and the stability boundary in a two-dimensional parameter plane which split the parameter plane into stable and unstable region. A focus is to understand the bifurcation diagram. We calculated the essential spectra of the wavetrains to understand the stability of the waves. Due to the solution crosses the stability boundary, disorderly act of predator and prey is found.
Where is the kernel function, denotes convolution and the parameters A,B,C and d are positive.
International Conference on Advances in Computational Mathematics
University of Dhaka, Dhaka, Bangladesh