Abstract
In this paper we derive a lattice model with innite distributed delay to describe the growth of a single-species population in a 2D patchy environment with innite number of patches connected locally by diusion and global interaction. We consider the existence of traveling wave solutions when the birth rate is large enough that each patch can sustain a positive equilibrium. When the birth function is monotone, we prove that there exists a traveling wave solution connecting two equilibria with wave speed c > c ( ) by using the monotone iterative method and super and subsolution technique, where 2 [0; 2 ] is any xed direction of propagation. When the birth function is
Highlights
In 1990, Aiello and Freedman [1] derived the following model to describe the growth of a single-species population: ui(t) = αum(t) − γui(t) − αe−γτ um(t − τ ), um(t) = αe−γτ um(t − τ ) − βu2m(t)
Β and γ are positive constants, um and ui denote the number of immature and mature members of the population, respectively; the delay τ > 0 is the time taken from birth to maturity. They assumed that the maturation delay τ is known exactly and that all individuals take this amount of time to mature
They showed that the unique positive equilibrium of (1.1) is globally asymptotically stable
Summary
For the model (1.5), its discrete version was firstly proposed by Weng et al [30] They considered the growth of a single-species population living in a patch environment consisting of all integer nodes of a 1D lattice. By the discrete Fourier transform, Weng et al [30] obtained the following lattice differential equations dwj (t) dt They established the spreading speed and the existence of monotone traveling waves for (1.7) when the birth function is monotone. Which models the growth of a single-species population with two age classes distributed over a patchy environment consisting of all integer nodes of a 2D lattice They studied the well-posedness of the initial-value problem and established the existence of monotone traveling waves for wave speed c ≥ c∗(θ) > 0, where θ is any fixed direction of propagation. The existence of non-trivial traveling waves of equation (2.7) is obtained in Section 3, where we consider two cases, namely, the monotone birth function and the non-monotone birth function
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