Permanence for Nicholson-type delay systems with nonlinear density-dependent mortality terms

Abstract

In this paper, we study the generalized Nicholson-type delay systems with nonlinear density-dependent mortality terms. Under proper conditions, we establish some criteria to guarantee the permanence of this model. Moreover, we give two examples to illustrate our main results.

Introduction

In a classic study of population dynamics, a delay differential equation model:

$$x^{\prime }\left(t\right)=-\delta x\left(t\right)+Px(t-\tau )e^{-ax(t-\tau )}\text{,}$$

is frequently used, where $x\left(t\right)$ is the size of the population at time $t,P$ is the maximum per capita daily egg production, $\frac{1}{a}$ is the size at which the population reproduces at its maximum rate, $\delta$ is the per capita daily adult death rate, and $\tau$ is the generation time. This equation was introduced by Nicholson [1] to model laboratory fly population. Its dynamics was later studied in [2], [3], where this model was referred to as the Nicholsons blowflies equation [2].

Recently, to describe the models of Marine Protected Areas and B-cell Chronic Lymphocytic Leukemia dynamics that belong to the Nicholson-type delay differential systems, Berezansky et al. [4] proposed the following Nicholson-type delay systems:

$$\{\begin{array}{c}{x}_1^{\prime }\left(t\right)=-a_1{x}_1\left(t\right)+b_1{x}_2\left(t\right)+c_1{x}_1(t-\tau )e^{-x_1(t-\tau )}\text{,}\hfill \\ x_2^{\prime }\left(t\right)=-a_2{x}_2\left(t\right)+b_2{x}_1\left(t\right)+c_2{x}_2(t-\tau )e^{-x_2(t-\tau )}\text{,}\hfill \end{array}$$

with initial conditions:

$$x_i\left(s\right)={\phi }_i\left(s\right)\text{,}s\in [-\tau ,0],{\phi }_i\left(0\right)>0\text{,}$$

where ${\phi }_i\in C([-\tau ,0],[0,+\infty )),a_i,b_i,c_i$ and $\tau$ are nonnegative constants, $i=1,2$.

According to the new studies in population dynamics (see [5], [6]), a linear model of density-dependent mortality will be most accurate for populations at low densities, and marine ecologists are currently in the process of constructing new fishery models with nonlinear density-dependent mortality rates. Therefore, Berezansky et al. [5] point out an open problem: How about the dynamic behaviors of the following scalar Nicholson’s blowflies model with a nonlinear density-dependent mortality term:

$$x^{\prime }\left(t\right)=-D\left(x\left(t\right)\right)+Px(t-\tau )e^{-x(t-\tau )}\text{,}$$

where the nonlinear density-dependent mortality term $D\left(x\right)$ might have one of the following forms: $D\left(x\right)=\frac{ax}{b+x}$ or $D\left(x\right)=a-b{e}^{-x}$ with constants $a,b>0$.

Now, a corresponding question arises: How about the dynamic behaviors of Nicholson-type delay differential systems with nonlinear density-dependent mortality terms. The main purpose of this paper is to give the conditions to ensure the permanence of Nicholson-type delay differential systems with nonlinear density-dependent mortality terms. Since the coefficients and delays in differential equations of population and ecology problems are usually time-varying in the real world, we will consider the following Nicholson-type delay systems with nonlinear density-dependent mortality terms:

$$\{\begin{array}{c}{x}_1^{\prime }\left(t\right)=-D_{11}(t,x_1\left(t\right))+D_{12}(t,x_2\left(t\right))+c_1\left(t\right)x_1(t-{\tau }_1\left(t\right))e^{-{\gamma }_1\left(t\right)x_1(t-{\tau }_1\left(t\right))}\text{,}\hfill \\ x_2^{\prime }\left(t\right)=-D_{22}(t,x_2\left(t\right))+D_{21}(t,x_1\left(t\right))+c_2\left(t\right)x_2(t-{\tau }_2\left(t\right))e^{-{\gamma }_2\left(t\right)x_2(t-{\tau }_2\left(t\right))}\text{,}\hfill \end{array}$$

where

$$D_{ij}(t,x)=\frac{a_{ij}\left(t\right)x}{b_{ij}\left(t\right)+x}\text{or}{D}_{ij}(t,x)=a_{ij}\left(t\right)-b_{ij}\left(t\right)e^{-x}\text{,}$$

$a_{ij},b_{ij},c_i,{\gamma }_i:R\to (0,+\infty )$ are all continuous functions bounded above and below by positive constants, ${\tau }_i:R\to [0,+\infty )$ is a bounded continuous function, $r_i={sup}_{t\in R}{\tau }_i\left(t\right)>0$, and $i,j=1,2$.

Throughout this paper, the set of all (nonnegative) real vectors will be denoted by $R\left(R_{+}\right)$. Let

$$C=C([-r_1,0],R)\times C([-r_2,0],R)$$

be the continuous functions space equipped with the usual supremum norm $\Vert \cdot \Vert$, and let

$$C_{+}=C([-r_1,0],R_{+})\times C([-r_2,0],R_{+})\text{.}$$

For the sake of convenience, we set

$$g^{+}=\underset{t\in R}{sup}g\left(t\right)\text{,}{g}^{-}=\underset{t\in R}{inf}g\left(t\right)\text{,}$$

where $g$ is a bounded continuous function defined on $R$.

If $x_i\left(t\right)$ is defined on $[t_0-r_i,\sigma )$ with $t_0,\sigma \in R$ and $i=1,2$, then we define $x_t\in C$ as $x_t=(x_t^1,x_t^2)$ where $x_t^i\left(\theta \right)=x_i(t+\theta )$ for all $\theta \in [-r_i,0]$ and $i=1,2$. For any $\phi ,\psi \in C$, we write $\phi \le \psi$ if $\psi -\phi \in C_{+},\phi <\psi$ if $\phi \le \psi$ and $\phi \ne \psi$.

Due to the biological interpretation of model (1.5), only positive solutions are meaningful and therefore admissible. Thus, we just consider the admissible initial conditions

$$x_{t_0}=\phi \text{,}\phi \in C_{+}\text{and}{\phi }_i\left(0\right)>0\text{,}i=1,2\text{.}$$

We write $x_t(t_0,\phi )\left(x(t;t_0,\phi )\right)$ for a solution of the initial value problems (1.5), (1.6). Also, let $[t_0,\eta \left(\phi \right))$ be the maximal right-interval of existence of $x_t(t_0,\phi )$.

The remaining part of this paper is organized as follows. In Sections 2 Permanence of Nicholson-type delay systems with, 3 Permanence of Nicholson-type delay systems with, we shall derive new sufficient conditions for checking the permanence of model (1.5). In Section 4, we shall give some examples and remarks to illustrate our results obtained in the previous sections.