Dynamics of a two-species stage-structured model incorporating state-dependent maturation delays

Abstract

This paper is devoted to a cooperative model composed of two species with stage structure and state-dependent maturation delays. Firstly, positivity and boundedness of solutions are addressed to describe the population survival and the natural restriction of limited resources. It is shown that for a given pair of positive initial functions, the two mature populations are uniformly bounded away from zero and that the two mature populations are bounded above only if the the coupling strength is small enough. Moreover, if the coupling strength is large enough then the two mature populations tend to infinity as the time tends to infinity. In particular, the positivity of the two immature populations has been established under some additional conditions. Secondly, the existence and patterns of equilibria are investigated by means of degree theory and Lyapunov-Schmidt reduction. Thirdly, the local stability of the equilibria is also discussed through a formal linearization. Fourthly, the global behavior of solutions is discussed and the explicit bounds for the eventual behaviors of the two mature populations and two immature populations are obtained. Finally, global asymptotical stability is investigated by using the comparison principle of the state-dependent delay equations.

Global Dynamics of a Predator-Prey Model with State-Dependent Maturation-Delay

“…From (19), we have 1 − bK k 2 dr [nbe −d j τ(0) − dk 1 ] > 0, and noting that ε > 0 can be arbitrarily small, then we have x = x. By (30) and let n, m → ∞, we get y = y.…”

“…The positive equilibrium (x * , y * ) in system (11) is globally asymptotically stable provided that nbe −d j τ(y * ) K 1+k 1 K > d and that the conditions ( 13), (19) hold true.…”

“…In 2017, Lv et al [14] studied an SDTD competitive model, where the SDTD is taken to be an increasing differentiable bounded function of the number of its own population, and completely analyzed the global stability of the equilibria by using the comparison principle and iterative method. For other non-predator-prey biological models with SDTD, we refer to [15,16,17,18,19] and the references therein.…”

Stability analysis of a stage‐structure model with spatial heterogeneity

“…Hence, age structure plays an important role in the modeling of multi-species populations interactions and has been widely studied. [8][9][10][11][12][13] Motivated by the works of Lam and Ni 1 and Liu et al, 14 and Yan and Guo, 15 we have investigated the following stage-structured Lotka-Volterra competition model in heterogeneous environments:…”

“…In the natural world, however, many species go through a number of stages (for example, immature and mature stages) during their lifetime, immature prey species are usually protected by mature prey, juvenile predators are not yet capable of predation, and some species take a long time from birth to adulthood. Hence, age structure plays an important role in the modeling of multi‐species populations interactions and has been widely studied 8–13 …”