Almost periodic solution of the non-autonomous two-species competitive model with stage structure. (English) Zbl 1163.34030
The system of differential equations
\[ \begin{cases} \dot{x}_1(t)=-a_1(t)x_1(t)+b_1(t)x_2(t),\\ \dot{x}_2(t)=a_2(t)x_2(t)-b_2(t)x_2(t)-c(t)x^2_2(t)-\beta_1(t)x_2(t)x_3(t),\\ \dot{x}_3(t)=x_3(t)(d(t)-e(t)x_3(t)-\beta_2(t)x_2(t)), \end{cases} \]
as a non-autonomous two-species competitive model with stage structure is investigated. The aim of this article is, by further developing the analytic technique of K. Gopalsamy [J. Aust. Math. Soc., Ser. B 27, 346–360 (1986; Zbl 0591.92022)] and by constructing a suitable Lyapunov function, to obtain a set of “easily verifiable” sufficient conditions to ensure the existence of a unique, globally attractive, positive , almost periodic solution of system (1). An example is presented.
\[ \begin{cases} \dot{x}_1(t)=-a_1(t)x_1(t)+b_1(t)x_2(t),\\ \dot{x}_2(t)=a_2(t)x_2(t)-b_2(t)x_2(t)-c(t)x^2_2(t)-\beta_1(t)x_2(t)x_3(t),\\ \dot{x}_3(t)=x_3(t)(d(t)-e(t)x_3(t)-\beta_2(t)x_2(t)), \end{cases} \]
as a non-autonomous two-species competitive model with stage structure is investigated. The aim of this article is, by further developing the analytic technique of K. Gopalsamy [J. Aust. Math. Soc., Ser. B 27, 346–360 (1986; Zbl 0591.92022)] and by constructing a suitable Lyapunov function, to obtain a set of “easily verifiable” sufficient conditions to ensure the existence of a unique, globally attractive, positive , almost periodic solution of system (1). An example is presented.
Reviewer: Tatiana Mamedova (Saransk)
MSC:
34C60 | Qualitative investigation and simulation of ordinary differential equation models |
92D25 | Population dynamics (general) |
34C27 | Almost and pseudo-almost periodic solutions to ordinary differential equations |
34D23 | Global stability of solutions to ordinary differential equations |
Keywords:
almost periodic solution; competition; stage structure; Lyapunov function; globally attractiveCitations:
Zbl 0591.92022References:
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