Stability in a Simple Food Chain System with Michaelis-Menten Functional Response and Nonlocal Delays

and Applied Analysis 3 Lemma 4 (see [21]). Let u(t, x) ∈ C([0,∞) × Ω) ∩ C 2,1 ((0,∞)×Ω) be the nontrivial positive solution of the system u t − dΔu = Bu (t − τ, x) ± A 1 u (t, x) − A 2 u 2 (t, x) , (t, x) ∈ (0,∞) × Ω, ∂u ∂] = 0, (t, x) ∈ (0,∞) × ∂Ω, u (t, x) = φ (t, x) ≥ 0, (t, x) ∈ (−τ, 0) × Ω, (6) where A 1 ≥ 0, B, A 2 , τ > 0. The following results hold: (1) if B±A 1 > 0, then u → (B ±A 1 )/A 2 uniformly onΩ as t → ∞; (2) if B±A 1 ≤ 0, then u → 0 uniformly onΩ as t → ∞. Throughout this paper, we assume that K i (x, y, t) = G i (x, y, t) k i (t) , x, y ∈ Ω, k i (t) ≥ 0; ∫ Ω G i (x, y, t) dx = ∫ Ω G i (x, y, t) dy = 1, t ≥ 0;


Introduction
The overall behavior of ecological systems continues to be of great interest to both applied mathematicians and ecologists.Two species predator-prey models have been extensively investigated in the literature.But recently more and more attention has been focused on systems with three or more trophic levels.For example, the predator-prey system for three species with Michaelis-Menten type functional response was studied by many authors [1][2][3][4].However, the systems in [1][2][3][4] are either with discrete delay or without delay or without diffusion.In view of individuals taking time to move, spatial dispersal was dealt with by introducing diffusion term to corresponding delayed ODE model in previous literatures, namely, adding a Laplacian term to the ODE model.In recent years, it has been recognized that there are modelling difficulties with this approach.The difficulty is that diffusion and time delay are independent of each other, since individuals have not been at the same point at previous times.Britton [5] made a first comprehensive attempt to address this difficulty by introducing a nonlocal delay; that is, the delay term involves a weighted-temporal average over the whole of the infinite domain and the whole of the previous times.
Motivated by the work above, we are concerned with the following food chain model with Michaelis-Menten type functional response: ) ,  2 (, )  2 +  2 (, ) ×  2 (, ,  − )   − 33  3 ) , (1) for  > 0,  ∈ Ω with homogeneous Neumann boundary conditions and initial conditions   (, ) =   (, ) ≥ 0 ( = 1, 2) , (, ) ∈ (−∞, 0] × Ω,  3 (0, ) =  3 () ≥ 0,  ∈ Ω, where   is bounded, Hölder continuous function and satisfies   /] = 0 ( = 1, 2, 3) on (−∞, 0] × Ω.Here, Ω is a bounded domain in R  with smooth boundary Ω and /] is the outward normal derivative on Ω.  (, ) represents the density of the th species (prey, predator, and top predator resp.) at time  and location  and thus only nonnegative   (, ) is of interest.The parameter  1 is the intrinsic growth rate of the prey, and  2 and  3 are the death rates of the predator and top-predator.  is the intraspecific competitive rate of the th species. 12 is the maximum predation rate. 23 and  32 are the efficiencies of food utilization of the predator and top predator, respectively.We assume the predator and top predator show the Michaelis-Menten (or Holling type II) functional response with  1 /( 1 +  1 ) and  2 /( 2 +  2 ), respectively, where  1 and  2 are half-saturation constants.For a through biological background of similar models, see [18,19].As our most knowledge, the tritrophic food chain model has been found to have many interesting biological properties, such as the coexistence and the Hopf bifurcation.However, the effect of nonlocal time delays on the coexistence has not been reported.Our paper mainly concerns this perspective.
The main purpose of this paper is to study the global asymptotic behavior of the solution of system (1)-(3).The preliminary results are presented in Section 2. Section 3 contains sufficient conditions for the global asymptotic behaviors of the equilibria of system (1)-( 3) by means of the Lyapunov functional.Numerical simulations are carried out to show the feasibility of the conditions in Theorems 8-10 in Section 4. Finally, a brief discussion is given to conclude this work.

Preliminary Results
In this section, we present several preliminary results that will be employed in the sequel.
The following result was obtained by the method of upper and lower solutions and the associated iterations in [21].

Global Stability
In this section, we study the asymptotic behavior of the equilibrium of system (1)-( 3).In the beginning, we show the existence and uniqueness of positive steady state.(12).Parameter values are listed in the example in Section 4.
Let us consider the following equations: A direct computation shows that the above equations have only one positive solution (V * 1 , V * 2 ) if and only if is satisfied.Taking  1 into account, we consider the equation Suppose that  1 and  2 hold.We take  3 as a parameter and consider the following system: When the parameter  3 is sufficiently small, the first two equations in ( 15) can be approximated by (12).Moreover, by continuously increasing the value of  3 ,  3 goes up, and meanwhile the intersection point between  1 and  2 also goes up.However,  3 goes up faster than  2 , while  1 keeps still.In other words, there exists a critical value  3 such that the intersection point lies in  3 , which implies that there is a unique positive solution  * ( * 1 ,  * 2 ,  * 3 ) to ( 15), or equivalently (1)-( 3).Lemma 7. Assume that  2 and  2 hold, and then the positive steady state  * of system (1)-( 3) is locally asymptotically stable.
So   > 0,   > 0,   > 0 and     −   > 0 for  ≥ 1 from the direct calculation.According to the Routh-Hurwitz criterion, the three roots  ,1 ,  ,2 ,  ,3 of   () = 0 all have negative real parts.By continuity of the roots with respect to   and Routh-Hurwitz criterion, we can conclude that there exists a positive constant  such that Consequently, the spectrum of , consisting only of eigenvalues, lies in {Re  ≤ −}.It is easy to see that E * is locally asymptotically stable and follows from Theorem 5.1.1 of [23].
Theorem 8. Assume that 2 and  2 hold, and the positive steady state  * of system (1)-( 3) with nontrivial initial function is globally asymptotically stable.
Theorem 10.Suppose that  5 and  6 hold, and then the semitrivial steady state  1 of system (1)-( 3) with non-trivial initial functions is globally asymptotically stable.
Proof.We study the stability of the semi-trivial solution  1 = (ũ 1 , 0, 0).Similarly, the equations in (1) can be written as Define Calculating the derivative of () along  1 , we get from (54) that Define  () =  () It is easy to see that (60) Then we get   > 0 ( = 1, 2, 3).Therefore we have lim uniformly on Ω. Next, we consider the asymptotic behavior of  2 (, ) and  3 (, ).For any  > 0, integrating (57) over [0, ] yields  () + where 3) for the constant   which is independent of .Now we consider the boundedness of ‖|∇ 2 |‖ 2 and ‖|∇ 3 |‖ 2 .From the Green's identity, we obtain In the end, we show that the trivial solution  0 is an unstable equilibrium.Similarly to the local stability to  * , we can get the characteristic equation of  0 as If  = 1, then  1 = 0.It is easy to see that this equation admits a positive solution  =  1 .According to Theorem 5.1 in [23], we have the following result.

Discussion
In this paper, we incorporate nonlocal delay into a threespecies food chain model with Michaelis-Menten functional response to represent a delay due to the gestation of the predator.The conditions, under which the spatial homogeneous equilibria are asymptotically stable, are given by using the Lyapunov functional.We now summarize the ecological meanings of our theoretical results.Firstly, the positive equilibrium  * of system (1)-(3) exists under the high birth rate of the prey ( 1 ) and low death rates ( 2 and  3 ) of predator and top predator. * is globally stable if the intraspecific competition  11 is neither too big nor too small and the maximum harvest rates  12 ,  23 are small enough.Secondly, the semi-trivial equilibrium  2 of system (1)-( 3) exists if the birth rate of the prey ( 1 ) is high, death rate of predator  2 is low, and the death rate ( 3 ) exceeds the conversion rate from predator to top predator ( 32 ). 2 is globally stable if the maximum harvest rates  12 ,  23 are small and the intra-specific competition  11 is neither too big nor too small.Thirdly, system (1)-(3) has only one semi-trivial equilibrium  1 when the death rate ( 2 ) exceeds the conversion rate from prey to predator ( 21 ). 1 is globally stable if intra-specific competitions ( 11 ,  22 , and  33 ) are strong.Finally,  0 is unstable and the non-stability of trivial equilibrium tells us that not all of the populations go to extinction.Furthermore, our main results imply that the nonlocal delay is harmless for stabilities of all non-negative steady states of system (1)- (3).
There are still many interesting and challenging problems with respect to system (1)-(3), for example, the permanence and stability of periodic solution or almost periodic solution.These problems are clearly worthy for further investigations.