Persistence and extinction of a nonautonomous switching single-species population model

doi:10.1016/j.aml.2019.106187

Applied Mathematics Letters

Volume 103, May 2020, 106187

https://doi.org/10.1016/j.aml.2019.106187 Get rights and content

Abstract

A novel nonautonomous switching single-species population model is proposed and investigated. The sufficient conditions for persistence and extinction of the solutions of the model are obtained. Moreover, corresponding results are given for periodic switching system.

Introduction

A population model is a type of mathematical model that is applied to the study of population dynamics. Since the pioneer work of Malthusian growth model [1], there has been a fair amount of work on modeling single-species population growth using discrete models [2], continuous models [3], impulsive models [4] and even stochastic models [5]. Continuous model is comprised of autonomous model and non-autonomous model, whereas non-autonomous model can more exactly describe the population growth than autonomous one since it incorporates the effects of a time-varying environment on the growth of a population.

In [6], a general nonautonomous single-species population model $\dot{x} = x f (t, x)$ was investigated, where growth function $f (t, x) : R_{+ 0}^{2} \to R$ is continuous and differentiable with respect to $x$ , here $R_{+ 0}^{2} = {(t, x) \in R^{2} : t \geq 0, x \geq 0}$ . Sufficient conditions for persistence and extinction of the population were obtained. However, in reality, the qualitative changes of the growth function form an essential aspect of the dynamics of the ecosystem. The change usually cannot be described by the continuous model owing to environmental switch, such as switching seasons (rainy season and dry season), switching sites (the north–south migration of migratory birds), and switching states. Therefore a nonautonomous counterpart of the fundamental growth equations with periodic switching time is introduced in this paper, namely $\dot{x} = x f_{i} (t, x), t \in (t_{i - 1}, t_{i}],$ in which $f_{i}$ represents the growth function of the population in different time interval, and ${t_{i}}$ are the switching times, and $0 \leq t_{0} < t_{1} < t_{2} < \dots$ . We assume that the switching rule satisfies $t_{i} - t_{i - 1} = w_{i}$ with $w_{i + q} = w_{i}$ , $i = 1, 2, \dots, q$ , where $q$ is the number of switching subsystems, and then $T = \sum_{i = 1}^{q} w_{i}$ (means $t_{i + q} = t_{i} + T$ ) is the switching period. The subject of this paper is to investigate the persistence and extinction of the population to improve the work of [6], [7].

Section snippets

Persistence and extinction for system (2.1)

For convenience, we rewrite the system (1.1) as $\dot{x} = x f_{i} (t, x), t \in (n T + t_{i - 1}, n T + t_{i}], n \in Z_{+}, i = 1, 2, \dots, q,$ where $Z_{+}$ is the set of nonnegative integers. Denote $f (t, x) = \{\begin{matrix} f_{1} (t, x), t \in (n T + t_{0}, n T + t_{1}], \\ f_{2} (t, x), t \in (n T + t_{1}, n T + t_{2}], \\ \dots \dots \\ f_{q} (t, x), t \in (n T + t_{q - 1}, (n + 1) T + t_{0}], \end{matrix} and D_{n i} = (n T + t_{i - 1}, n T + t_{i}] \times R_{+ 0},$ and give following assumptions for all $n \in Z_{+}$ and $i = 1, 2, \dots, q$ .

$A1.$ The function $f_{i}$ is continuous and differentiable with respect to $x$ on $D_{n i}$ , and $\partial f_{i} ∕ \partial x$ is continuous on $D_{n i}$ .

$A2.$ The function $\partial f_{i} ∕ \partial x \leq 0$ on $D_{n i}$ .

$A3.$ There exist constants $β \geq 0$ and $k > 0$ ,

Persistence and extinction for periodic switching system

We now consider the special case in which $f_{i}$ is periodic in $t$ with period $T$ , that is,

$A5$ . The function $f_{i}$ satisfies $f_{i} (t + T, x) = f_{i} (t, x)$ for $t \in (n T + t_{i - 1}, n T + t_{i}]$ and $x \geq 0$ .

At the same time, the system (1.1) or (2.1) becomes the periodic switching system, and assumptions $A_{3}$ (ii), $A_{4}$ (ii) and $A_{3}^{0}$ (ii) can be simplified as the following forms:

$A_{3}^{*}$ (ii). There exists constant $k > 0$ , such that $\sum_{i = 1}^{q} \int_{t_{i - 1}}^{t_{i}} f_{i} (s, k) d s \leq 0$ .

$A_{4}^{*}$ (ii). There exists constant $0 < δ \leq k$ such that $\sum_{i = 1}^{q} \int_{t_{i - 1}}^{t_{i}} f_{i} (s, δ) d s \geq 0$ .

$A_{3}^{0 *}$ (ii). $\sum_{i = 1}^{q} \int_{t_{i - 1}}^{t_{i}} f_{i} (s, 0)$

CRediT authorship contribution statement

Yan Xu: Writing - original draft. Shujing Gao: Conceptualization, Methodology. Di Chen: Formal analysis.

References (7)

YuX. et al.
Persistence and ergodicity of a stochastic single species model with Allee effect under regime switching
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An Essay on the Principle of Population (1798) and a Summary View of the Principle of Population (1830)
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BulmerM.G.
Periodical insects
Amer. Nat.
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There are more references available in the full text version of this article.

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^☆: The research of S. Gao has been supported by the Natural Science Foundation of China (11961003, 11561004) and The Natural Science Foundation of Jiangxi Province, China (20192BCBL20004).

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Persistence and extinction of a nonautonomous switching single-species population model☆

Abstract

Introduction

Section snippets

Persistence and extinction for system (2.1)

Persistence and extinction for periodic switching system

CRediT authorship contribution statement

Commun. Nonlinear Sci.

An Essay on the Principle of Population (1798) and a Summary View of the Principle of Population (1830)

Periodical insects

Amer. Nat.