Permanence of a Semi-Ratio-Dependent Predator-Prey System with Nonmonotonic Functional Response and Time Delay

Abstract

Sufficient conditions for permanence of a semi-ratio-dependent predator-prey system with nonmonotonic functional response and time delay 𝑥̇₁(𝑡)=𝑥₁(𝑡)[𝑟₁(𝑡)−𝑎₁₁(𝑡)𝑥₁(𝑡−𝜏(𝑡))−𝑎₁₂(𝑡)𝑥₂(𝑡)/(𝑚²+𝑥²₁(𝑡))],  𝑥̇₂(𝑡)=𝑥₂(𝑡)[𝑟₂(𝑡)−𝑎₂₁(𝑡)𝑥₂(𝑡)/𝑥₁(𝑡)], are obtained, where 𝑥₁(𝑡) and 𝑥₂(𝑡) stand for the density of the prey and the predator, respectively, and 𝑚≠0 is a constant. 𝜏(𝑡)≥0 stands for the time delays due to negative feedback of the prey population.

1. Introduction

The dynamic relationship between predators and their preys has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance [1]. These problems may appear to be simple mathematically at first sight, but they are, in fact, often very challenging and complicated [2, 3].

Recently, Ding et al. [4] studied dynamics of a semi-ratio-dependent predator-prey system with the nonmonotonic functional response and delay 𝑥̇₁(𝑡)=𝑥₁(𝑡)[𝑟₁(𝑡)−𝑎₁₁(𝑡)𝑥₁𝑎(𝑡−𝜏(𝑡))−¹²(𝑡)𝑥₂(𝑡)𝑚²+𝑥²₁𝑥̇(𝑡)],₂(𝑡)=𝑥₂(𝑡)[𝑟₂𝑎(𝑡)−²¹(𝑡)𝑥₂(𝑡)𝑥₁(𝑡)], (1.1) with initial conditions 𝑥_𝑖(𝜃)=𝜙_𝑖(𝜃),𝜃∈[−𝜏^𝑢,0],𝜙_𝑖(0)>0,𝜙_𝑖∈𝐶([−𝜏^𝑢,0],𝑅₊),𝑖=1,2, (1.2) where 𝑥₁(𝑡) and 𝑥₂(𝑡) stand for the density of the prey and the predator, respectively, and 𝑚≠0 is a constant. 𝜏(𝑡)≥0 stands for the time delays due to negative feedback of the prey population. 𝑟₁(𝑡),𝑟₂(𝑡) stand for the intrinsic growth rates of the prey and the predator, respectively. 𝑎₁₁(𝑡) is the intraspecific competition rate of the prey. 𝑎₁₂(𝑡) is the capturing rate of the predator. The predator grows with the carrying capacity 𝑥(𝑡)/𝑎₂₁(𝑡) proportional to the population size of the prey or prey abundance. 𝑎₂₁(𝑡) is a measure of the food quality that the prey provided for conversion into predator birth. Assumed that 𝑟_𝑖(𝑡),𝑎_𝑖𝑗(𝑡),𝑖,𝑗=1,2, are continuously positive periodic functions with period 𝜔, by using the continuation theorem of coincidence degree theory, the existence of a positive periodic solution for the semi-ratio-dependent predator-prey system with nonmonotonic functional responses and time delay is established. For the ecological sense of the system (1.1) we refer to [5–8] and the references cited therein.

As we know, permanence is one of the most important topics on the study of population dynamics. One of the most interesting questions in mathematical biology concerns the survival of species in ecological models. Biologically, when a system of interacting species is persistent in a suitable sense, it means that all the species survive in the long term. It is reasonable to ask for conditions under which the system is permanent. However, Ding et al. [4] did not investigate this property of the system (1.1).

Motivated by the above question, we will consider the permanence of the system (1.1). Unlike the assumptions of Ding et al. [4], we argue that a general nonautonomous nonperiodic system is more appropriate, and thus, we assume that the coefficients of system (1.1) satisfy: (A) 𝑟_𝑖(𝑡),𝑎_𝑖𝑗(𝑡),𝜏(𝑡),𝑖,𝑗=1,2, are nonnegative functions bounded above and below by positive constants.

Throughout this paper, for a continuous function 𝑔(𝑡), we set 𝑔^𝑙=inf_𝑡∈𝑅𝑔(𝑡),𝑔^𝑢=sup_𝑡∈𝑅𝑔(𝑡). (1.3)

It is easy to verify that solutions of system (1.1) corresponding to initial conditions (1.2) are defined on [0,+∞) and remain positive for all 𝑡≥0. In this paper, the solution of system (1.1) satisfying initial conditions (1.2) is said to be positive.

The aim of this paper is, by using the differential inequality theory, to obtain a set of sufficient conditions to ensure the permanence of the system (1.1).

2. Permanence

In this section, we establish a permanence result for system (1.1).

Lemma 2.1 (see [9]). If 𝑎>0,𝑏>0 and 𝑥̇≥𝑥(𝑏−𝑎𝑥), when 𝑡≥0 and 𝑥(0)>0, one has: lim_𝑡→+∞𝑏inf𝑥(𝑡)≥𝑎. (2.1) If 𝑎>0,𝑏>0 and 𝑥̇≤𝑥(𝑏−𝑎𝑥), when 𝑡≥0 and 𝑥(0)>0, one has: lim_𝑡→+∞𝑏sup𝑥(𝑡)≤𝑎. (2.2)

Proposition 2.2. Let (𝑥₁(𝑡),𝑥₂(𝑡)) be any positive solution of system (1.1), then lim_𝑛→+∞sup𝑥_𝑖(𝑛)≤𝑀_𝑖,𝑖=1,2, (2.3) where 𝑀₁=𝑟^𝑢₁𝑎^𝑙₁₁exp{−𝑟^𝑢₁𝜏^𝑢},𝑀₂=𝑟^𝑢₂𝑀₁𝑎^𝑙₂₁. (2.4)

Proof. Let (𝑥₁(𝑡),𝑥₂(𝑡)) be any positive solution of system (1.1), from the first equation of system (1.1) one has 𝑥̇₁(𝑡)≤𝑟₁(𝑡)𝑥₁(𝑡). (2.5) By integrating both sides of the above inequality from 𝑡−𝜏(𝑡) to 𝑡 with respect to 𝑡, we obtain 𝑥₁(𝑡−𝜏(𝑡))≥𝑥₁(𝑡)exp{^𝑡_{𝑡−𝜏(𝑡)}−𝑟₁(𝑠)𝑑𝑠}≥𝑥₁(𝑡)exp{−𝑟^𝑢₁𝜏^𝑢}. (2.6) By substituting (2.6) into the first equation of system (1.1), one has 𝑥̇₁(𝑡)≤𝑥₁(𝑡)[𝑟₁(𝑡)−𝑎₁₁(𝑡)exp{−𝑟^𝑢₁𝜏^𝑢}𝑥₁(𝑡)]≤𝑥₁(𝑡)[𝑟^𝑢₁−𝑎^𝑙₁₁exp{−𝑟^𝑢₁𝜏^𝑢}𝑥₁(𝑡)]. (2.7) By Lemma 2.1, according to (2.7), it immediately follows that lim_𝑡→+∞sup𝑥₁𝑟(𝑡)≤^𝑢1𝑎^𝑙₁₁exp{−𝑟^𝑢₁𝜏^𝑢}∶=𝑀₁. (2.8) It follows that for any small positive constant 𝜀>0, there exists a 𝑇₁>0 such that 𝑥₁(𝑡)≤𝑀₁+𝜀,𝑡>𝑇₁. (2.9)
By substituting (2.9) into the second equation of system (1.1), one has
𝑥̇₂(𝑡)≤𝑥₂(𝑡)[𝑟₂𝑎(𝑡)−²¹(𝑡)𝑥₂(𝑡)𝑀₁+𝜀]≤𝑥₂(𝑡)[𝑟^𝑢₂−𝑎^𝑙₂₁𝑀₁𝑥+𝜀²(𝑡)]. (2.10) By Lemma 2.1, according to (2.10), we get lim_𝑡→+∞sup𝑥₂𝑟(𝑡)≤^𝑢2(𝑀₁+𝜀)𝑎^𝑙₂₁. (2.11) Setting 𝜀→0 yields that lim_𝑡→+∞sup𝑥₂𝑟(𝑡)≤^𝑢2𝑀₁𝑎^𝑙₂₁∶=𝑀₂. (2.12)
This completes the proof of Proposition 2.2.

Now we are in the position of stating the permanence of the system (1.1).

Theorem 2.3. Assume that 𝑟^𝑙₁−𝑎^𝑢₁₂𝑀₂/𝑚²>0 hold, then system (1.1) is permanent, that is, there exist positive constants 𝑚_𝑖,𝑀_𝑖,𝑖=1,2, which are independent of the solutions of system (1.1), such that for any positive solution (𝑥₁(𝑡),𝑥₂(𝑡)) of system (1.1) with initial condition (1.2), one has 𝑚_𝑖≤lim_𝑡→+∞inf𝑥_𝑖(𝑡)≤lim_𝑡→+∞sup𝑥_𝑖(𝑡)≤𝑀_𝑖,𝑖=1,2. (2.13)

Proof. By applying Proposition 2.2, we see that to end the proof of Theorem 2.3, it is enough to show that under the conditions of Theorem 2.3, lim_𝑡→+∞inf𝑥_𝑖(𝑡)≥𝑚_𝑖. (2.14) From Proposition 2.2, forall𝜀>0, there exists a 𝑇₂>0,forall𝑡>𝑇₂, 𝑥_𝑖(𝑡)≤𝑀_𝑖+𝜀. (2.15) By substituting (2.15) into the first equation of system (1.1), it follows that 𝑥̇₁(𝑡)≥𝑥₁(𝑡)[𝑟₁(𝑡)−𝑎₁₁(𝑡)(𝑀₁𝑎+𝜀)−¹²(𝑡)(𝑀₂+𝜀)𝑚²]. (2.16) By integrating both sides of the above inequality from 𝑡−𝜏(𝑡) to 𝑡 with respect to 𝑡, we obtain 𝑥₁(𝑡−𝜏(𝑡))≤𝑥₁(𝑡)exp{^𝑡_{𝑡−𝜏(𝑡)}−[𝑟₁(𝑠)−𝑎₁₁(𝑠)(𝑀₁𝑎+𝜀)−¹²(𝑠)(𝑀₂+𝜀)𝑚²]𝑑𝑠}≤𝑥₁(𝑡)exp{[−𝑟^𝑙₁𝜏^𝑙+𝑎^𝑢₁₁(𝑀₁+𝜀)𝜏^𝑢+𝑎^𝑢₁₂(𝑀₂+𝜀)𝜏^𝑢𝑚²]}. (2.17) By substituting the above inequality into the first equation of system (1.1), one has 𝑥̇₁(𝑡)≥𝑥₁(𝑡){𝑟₁𝑎(𝑡)−¹²(𝑡)(𝑀₂+𝜀)𝑚²−𝑎₁₁(𝑡)exp{[−𝑟^𝑙₁𝜏^𝑙+𝑎^𝑢₁₁(𝑀₁+𝜀)𝜏^𝑢+𝑎^𝑢₁₂(𝑀₂+𝜀)𝜏^𝑢𝑚²]}𝑥₁(𝑡)}≥𝑥₁(𝑡){𝑟^𝑙₁−𝑎^𝑢₁₂(𝑀₂+𝜀)𝑚²−𝑎^𝑢₁₁exp{[−𝑟^𝑙₁𝜏^𝑙+𝑎^𝑢₁₁(𝑀₁+𝜀)𝜏^𝑢+𝑎^𝑢₁₂(𝑀₂+𝜀)𝜏^𝑢𝑚²]}𝑥₁(𝑡)}. (2.18) By Lemma 2.1, under the conditions of Theorem 2.3, it immediately follows that lim_𝑡→+∞inf𝑥₁𝑟(𝑡)≥^𝑙1−𝑎^𝑢₁₂(𝑀₂+𝜀)/𝑚²𝑎^𝑢₁₁exp{[−𝑟^𝑙₁𝜏^𝑙+𝑎^𝑢₁₁(𝑀₁+𝜀)𝜏^𝑢+𝑎^𝑢₁₂(𝑀₂+𝜀)𝜏^𝑢/𝑚².]} (2.19) Setting 𝜀→0, yields that lim_𝑡→+∞inf𝑥₁𝑟(𝑡)≥^𝑙1−𝑎^𝑢₁₂𝑀₂/𝑚²𝑎^𝑢₁₁exp{[−𝑟^𝑙₁𝜏^𝑙+𝑎^𝑢₁₁𝑀₁𝜏^𝑢+𝑎^𝑢₁₂𝑀₂𝜏^𝑢/𝑚²]}∶=𝑚₁. (2.20) It follows that for the above positive constant 𝜀>0, there exists a 𝑇₂>0 such that 𝑥₁(𝑡)≥𝑚₁−𝜀,𝑡>𝑇₂. (2.21) By substituting (2.21) into the second equation of system (1.1), one has 𝑥̇₂(𝑡)≥𝑥₂(𝑡)[𝑟₂𝑎(𝑡)−²¹(𝑡)𝑥₂(𝑡)𝑚₁−𝜀]≥𝑥₂(𝑡)[𝑟^𝑙₂−𝑎^𝑢₂₁𝑚₁𝑥−𝜀²(𝑡)]. (2.22) By Lemma 2.1, according to (2.22), it immediately follows that lim_𝑡→+∞inf𝑥₂𝑟(𝑡)≥^𝑙2(𝑚₁−𝜀)𝑎^𝑢₂₁. (2.23) Setting 𝜀→0 yields that lim_𝑡→+∞inf𝑥₂𝑟(𝑡)≥^𝑙2𝑚₁𝑎^𝑢₂₁∶=𝑚₂. (2.24) This completes the proof of Theorem 2.3.