# American Institute of Mathematical Sciences

June  2017, 22(4): 1393-1423. doi: 10.3934/dcdsb.2017067

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

 College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China

* Corresponding author: S. Guo

Received  May 2016 Revised  November 2016 Published  February 2017

Fund Project: The second author is supported by NSF of China (Grants No. 11671123 & 11271115)

This paper is devoted to a cooperative model composed of two species withstage 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.

Citation: Shangzhi Li, Shangjiang Guo. Dynamics of a two-species stage-structured model incorporating state-dependent maturation delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1393-1423. doi: 10.3934/dcdsb.2017067
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##### References:
Simulations of system (3) illustrate that the synchronous equilibrium is globally asymptotically stable, where $\alpha=2,\gamma=0.1,\mu=0.1,\beta=0.365$
Simulations of system (3) illustrate that the synchronous equilibrium is globally asymptotically stable, where $\alpha=1.5,\gamma=0.2,\mu=0.1,\beta=0.365$
Simulations of system (3) illustrate that every solution of (3) is asymptotically synchronous and tends to infinity as $t$ tends to infinity, where $\alpha=2,\gamma=0.1,\mu=0.4,\beta=0.365$
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