A toxin-mediated size-structured population model: Finite difference approximation and well-posedness.

Hao Wang,Qihua Huang

doi:10.3934/mbe.2016015

Abstract

The question of the effects of environmental toxins on ecological communities is of great interest from both environmental and conservational points of view. Mathematical models have been applied increasingly to predict the effects of toxins on a variety of ecological processes. Motivated by the fact that individuals with different sizes may have different sensitivities to toxins, we develop a toxin-mediated size-structured model which is given by a system of first order fully nonlinear partial differential equations (PDEs). It is very possible that this work represents the first derivation of a PDE model in the area of ecotoxicology. To solve the model, an explicit finite difference approximation to this PDE system is developed. Existence-uniqueness of the weak solution to the model is established and convergence of the finite difference approximation to this unique solution is proved. Numerical examples are provided by numerically solving the PDE model using the finite difference scheme.

Highlights

The question of the effects of anthropogenic and natural environmental toxins on ecological communities is of great interest from both environmental and conservational points of view
Motivated by the fact that depending on age, weight, and/or size, different individuals may have different sensitivities to toxins, we developed a toxin-mediated size-structured model that is given by a system of first order fully nonlinear hyperbolic partial differential equations (PDEs)
It is very likely that this model is the first derivation of PDE model in the area of ecotoxicology

Summary

Introduction

The question of the effects of anthropogenic and natural environmental toxins on ecological communities is of great interest from both environmental and conservational points of view. We denote by ukj and vjk, the finite difference approximation of u(xj, tk) and v(xj, tk), respectively, and let gjk = g(xj, P (tk)), μkj = μ(xj, P (tk), v(xj, tk)), βjk = β(xj, P (tk), v(xj, tk)), akj = a(xj, tk), σjk = σ(xj, tk), Ek = E(tk).

Results

Conclusion