Analysis of a Delay Nonautonomous Predator-prey System with Disease in the Prey

In this paper we have considered a nonautonomous predator-prey model with time delay due to gestation, in which a disease that can be transmitted by contact spreads among the prey only. Here, we have established some sufficient conditions on the permanence of the system by using inequality analytical technique. By Lyapunov functional method, we have also obtained some sufficient conditions for global asymptotic stability of this model. We have observed that the time delay has no effect on the permanence of the system but it has an effect on the global asymptotic stability of this model. The aim of the analysis of this model is to identify the parameters of interest for further study, with a view to informing and assisting policy-makers in targeting prevention and treatment resources for maximum effectiveness.


Introduction
The mathematical epidemic models have received much attention from researchers after the pioneering work of Kermack-McKendrick [1] on SIRS (susceptible-infectiveremoval-susceptible) systems, in which the evolution of a disease which gets transmitted upon contact is described.Ecology and epidemiology are two major and distinct fields of research in their own right.Lotka [2] and Volterra [3] established seminal works on the mathematical modelling of predator-prey and competing species in terms of simultaneous non-linear differential equations, making the first breakthrough in modern mathematical ecology.Ecoepidemiology is the study of interacting species in which a disease spreads.The study of Ecoepidemiology has important ecological significance.Ecoepidemiology research is becoming important as it involves persistence-extinction threshold of each population in systems of two or more interacting species subjected to parasitism [4][5][6][7][8].In the natural world, however, species do not exist alone while species spreads disease, it also competes with the other species for space or food, or is predated by other species.Understanding how parasites affect biodiversity and ecosystem dynamics is an important question in conservation biology as infectious disease can be a factor in regulating host population.On the other hand, successful invasion of a parasite into a host population and resulting host-parasite dynamics can depend essentially on other members of a host's community such as predators.Predation can dramatically shape the structures of community and ecosystem and becomes particularly interesting in host-parasite systems because predation itself can strongly alter dynamics of hosts and parasites.Predators may also prevent proper invasion of parasites into host population.Therefore, it is very significant biologically to consider the effect of interacting species when we study the dynamical behaviours of epidemiological systems.Scientists have paid lots of attention to merge these two important areas of research [4][5][6][7][8][9][10][11][12][13].
Nonautonomous phenomenon often occurs in many realistic ecoepidemiological models.The nonautonomous phenomenon occurs mainly due to the seasonal variations, which make the population behave periodically.Pathogen contact rate and infectivity vary seasonally and is generally larger in spring and autumn than in summer and winter.This seasonal pattern is related to moderate temperatures in spring and autumn which improves pathogen survival and favors high insect activity.Since biological and environmental parameters are naturally subject to fluctuation in time, the effects of a periodically varying environment are considered as important selective forces on systems in a fluctuating environment.To investigate this kind of phenomenon, in the model, the coefficients should be periodic functions, then the system is called periodic system.The nonautonomous ecoepidemiological models can be regarded as an extension of the periodic ecoepidemiological models.Therefore, the research on the nonautonomous ecoepidemiological dynamical models is also very important.
Considering the above facts, in this paper we have considered a nonautonomous predator-prey model with time delay due to gestation, in which a disease that can be transmitted by contact spreads among the prey only.In the proposed system, all the coefficients are time-dependent, i.e., it is nonautonomous.Usually, such systems do not have any disease-free equilibrium and endemic equilibrium.There are many methods to deal with autonomous systems, but they may not be suitable to nonautonomous systems.Therefore, it is more difficult to study the dynamical behaviours in nonautonomous case.Here, we have established some sufficient conditions on the permanence (uniformly persistent) of the system by using inequality analytical technique.By Lyapunov functional method, we have also obtained some sufficient conditions for the global asymptotic stability of this system.

The basic mathematical model
Here, we have considered an ecoepidemiological system consisting of three species, namely, the sound prey (which is susceptible), the infected prey (which becomes infective by some viruses) and the predator population.Our mathematical model is formulated as the following system of nonautonomous delay differential equations: Here, (x 1 (t), x 2 (t), x 3 (t)) are the densities of the sound prey (which is susceptible), the infected prey (which becomes infective by some viruses) and the predator population, respectively at time t.
The model is derived under following assumptions.The quantities (A1) We assume that only susceptible prey x 1 (t) is capable of reproducing with logistic law.The mortality terms for susceptible and infected prey are of density dependence [14], say, the rate k 1 (t) for the susceptible prey and k 2 (t) for the infected prey.The infected prey x 2 (t) is removed by death or by predation before having the possibility of reproducing.However, the infected prey population x 2 (t) still contributes with x 1 (t) to population growth toward the carrying capacity.
(A2) We assume that the disease spreads among the prey population only and the disease is not genetically inherited.The infected prey population does not recover nor becomes immune.The incidence is assumed to be the simple mass action form β(t)x 1 (t)x 2 (t), where β(t) is called the transmission rate function.
(A3) The predators hunt with possibly different predation rate functions a 1 (t) and a 2 (t) on susceptible and infected preys respectively.This is in accordance with the fact that the infected individuals can be caught more easily.For example, Peterson and Page [15] have indicated that wolf attacks on moose are more often successful if the moose is heavily infected by "Echinococcus granucosus".c 1 (t) and c 2 (t) denote the growth rate functions of the predator due to predation on susceptible and infected preys respectively; and the rate of growth are respectively of the forms: τ > 0 is the fixed time delay due to gestation lag.
The initial conditions of (1) are given as where For a biological meaning, we further assume that ϕ i (0) > 0, i = 1, 2, 3.
If not, then there exists a t 1 > 0, such that x 3 (t 1 ) = 0, and Then which is a contradiction.Hence, x 3 (t) > 0, for all t ≥ 0. This completes the proof.
3 Permanence of system (1) Here we wish to discuss the permanence (uniformly persistent) of system (1) with initial conditions (2), which demonstrates how this system will be uniformly persistent, this means that the long-term survival (i.e., will not vanish in time) of all components of the system (1) with initial conditions (2), under some conditions.Let f l = inf t≥0 f (t), f u = sup t≥0 f (t), for a continuous and bounded function f (t) defined on [0, +∞).

Definition 1.
System ( 1) is said to be uniformly persistent, i.e., the long-term survival (will not vanish in time) of all components of the system (1), if there are positive constants v i and w i (i = 1, 2, 3) such that: hold for any solution (x 1 (t), x 2 (t), x 3 (t)) of ( 1) with initial conditions (2).Here v i and w i (i = 1, 2, 3) are independent of (2).
From (1b) we have, By condition (3) and Theorem 2, we conclude that since M 3 can be chosen sufficiently close to M * .This completes the proof.

Global asymptotic stability
In this section, we derive sufficient conditions for global asymptotic stability of system (1) with initial conditions of type (2).We now state a definition of global asymptotic stability of solutions of system (1).Definition 2. System (1) with initial conditions ( 2) is said to be globally asymptotically stable if hold for any two solutions (x 1 (t), x 2 (t), x 3 (t)) and (u 1 (t), u 2 (t), u 3 (t)) of ( 1) with initial conditions of type (2).
∩ [e j , f j ] = φ and f i − e i = f j − e j > 0, for all i, j = 1, 2, . . .and i = j, one has ∞ k=1 f k e k B i (t)dt = ∞, then system (1) with initial conditions (2) is globally asymptotically stable.Here, where m, M are given in Theorem 4.
Then the right-upper derivative of V 1 (t), V 2 (t) and V 3 (t) along the solution of system (1) and ( 2) are given below: Define Calculating the right-upper derivative of V 4 (t) along the solution of system (1) and ( 2), we have ), then by using ( 15)-( 19), we have where T 6 is defined in Theorem 4 and B i (t), (i = 1, 2, 3) are defined in (14).
Integrating (20) from T 6 to t, we have By assumptions ( 14) about B i (t), (i = 1, 2, 3) and the boundedness of (x This shows that system (1) with initial conditions (2) is globally asymptotically stable.This completes the proof.
Corollary 2. If system (1) is ω-periodic and conditions in Theorems 4 and 5 are valid, then there exists a unique positive ω-periodic solution which is globally asymptotically stable.

Conclusions
In this paper we have considered a nonautonomous predator-prey model with time delay due to gestation, in which a disease that can be transmitted by contact spreads among the prey only.The most basic and important questions to ask for biological systems in the theory of mathematical epidemiology are the persistence, extinctions, the existence of periodic solutions, global stability, etc. [1,[19][20][21][22][23].Here, we have established some sufficient conditions for the permanence (uniformly persistent) of the above system by using inequality analytical technique.By Lyapunov functional method, we have also obtained some sufficient conditions for the global asymptotic stability of this model.We have observed that the time delay has no effect on the permanence of the system but it has an effect on the global asymptotic stability of this model.The aim of the analysis of this model is to identify the parameters of interest for further study, with a view to informing and assisting policy-makers in targeting prevention and treatment resources for maximum effectiveness.