Stability and Traveling Fronts in Lotka-Volterra Competition Models with Stage Structure

Abstract

This paper is concerned with a delay differential equation model for the interaction between two species, the adult members of which are in competition. The competitive effects are of the Lotka-Volterra kind, and in the absence of competition it is assumed that each species evolves according to the predictions of a simple age-structured model which reduces to a single equation for the total adult population. For each of the two species the model incorporates a time delay which represents the time from birth to maturity of that species. Thus, the time delays appear in the adult recruitment terms.The dynamics of the model are determined, and global stability results are established for each equilibrium. The equilibria of the model involve the maturation delays. The criteria for global convergence to each equilibrium are sharp and involve these delays.A reaction-diffusion extension of the model is also studied for the case when only the adult members of each species can diffuse. We prove the existence of a traveling front solution connecting the two boundary equilibria for the case when there is no coexistence equilibrium. This represents invasion by the stronger species of territory previously inhabited only by the weaker. The proof of the existence of such a front uses Wu and Zou's theory for traveling front solutions of delayed reaction-diffusion systems.

Diffusive and nondiffusive population models

“…Time delay may be used to model immature and mature stages; see [2] for an example. For the analysis of a system of delayed equations, we refer to [4]. More references can be found in the monograph of Kuang [81].…”

Hopf Bifurcation and Stability of a Competition Diffusion System with Distributed Delay

“…Delays are often incorporated into population models for resource regeneration times, maturing times or gestation periods [6]. Gourley et al [1,7] considered reaction diffusion systems with delay terms. The system represents two interacting species for x ∈ (−∞, +∞), t ≥ 0, and the double convolution is defined by…”

“…Gourley et al [1,7] presented a computational method for determining regions in parameter space corresponding to linear instability of a spatially uniform steady state solution. Similarly, for the variety of kernels, Beretta et al [2], Hasting [10], Yamada et al [14] discussed the corresponding models.…”

A stage-structured delayed reaction-diffusion model for competition between two species.

“…Then using the fluctuation lemma and constructing sequences approaching equilibrium points we showed that the global stability of the equilibria in our model can be completely determined. The mathematical methods used in our proofs are inspired by AlOmari and Gourley's work [63]. By the study of the non-spatial system (3.1), we conclude that (i) the two interactive species with stage structure can persist in a stream; (ii) one species out-competes the other one, and the species cannot coexist; (iii) the two species can coexist and approach a stable population density in long term under certain conditions (i.e.…”

“…where There have been a number of investigations of traveling wave solutions and asymptotic behavior in terms of spreading speeds for various evolution systems including nonlinear reaction-diffusion systems [54,55,56] For a delayed Lotka-Volterra type competition model, Al-Omari and Gourley [63] showed that for c ≥c, wherec is some number, the system has a nondecreasing traveling wave solution connecting two mono-culture equilibria. Liang and…”