Nonlinear Analysis: Real World Applications 7 6 4 8 www.elsevier.com/locate/na Analysis of a predator prey model with modified Leslie Gower Holling-type II schemes with time delay A.F. Nindjin a, M.A. Aziz-Alaoui b,, M. Cadivel b a Laboratoire de Mathématiques Appliquées, Université de Cocody, BP 58, Abidjan, Côte d Ivoire, France b Laboratoire de Mathématiques Appliquées, Université du Havre, 5 rue Philippe Lebon, B.P. 54, 7658 Le Havre Cedex, France Received 7 July 5; accepted 7 October 5 Abstract Two-dimensional delayed continuous time dynamical system modeling a predator prey food chain, based on a modified version of Holling type-ii scheme is investigated. By constructing a Liapunov function, we obtain a sufficient condition for global stability of the positive equilibrium. We also present some related qualitative results for this system. 5 Elsevier Ltd. All rights reserved. Keywords: Time delay; Boundedness; Permanent; Local stability; Global stability; Liapunov functional. Introduction The dynamic relationship between predators their prey has long been will continue to be one of dominant themes in both ecology mathematical ecology due to its universal existence importance. A major trend in theoretical work on prey predator dynamics has been to derive more realistic models, trying to keep to maximum the unavoidable increase in complexity of their mathematics. In this optic, recently [], see also [,5,6] has proposed a first study of two-dimensional system of autonomous differential equation modeling a predator prey system. This model incorporates a modified version of Leslie Gower functional response as well as that of the Holling-type II. They consider the following model ẋ = a bx ẏ = a c y x c y x y x, with the initial conditions x> y>. This two species food chain model describes a prey population x which serves as food for a predator y. The model parameters a, a, b, c, c, are assuming only positive values. These parameters are defined as follows: a is the growth rate of prey x, b measures the strength of competition among individuals of species x, c Corresponding author. Tel./fax: 33 3 74 4. E-mail address: Aziz-Alaoui@univ-lehavre.fr M.A. Aziz-Alaoui. 468-8/$ - see front matter 5 Elsevier Ltd. All rights reserved. doi:.6/j.nonrwa.5..3
A.F. Nindjin et al. / Nonlinear Analysis: Real World Applications 7 6 4 8 5 is the maximum value of the per capita reduction rate of x due to y, respectively, measures the extent to which environment provides protection to prey x respectively, to the predator y, a describes the growth rate of y, c has a similar meaning to c. It was first motivated more by the mathematics analysis interest than by its realism as a model of any particular natural dynamical system. However, there may be situations in which the interaction between species is modelized by systems with such a functional response. It may, for example, be considered as a representation of an insect pest spider food chain. Furthermore, it is a first step towards a predator prey model of Holling Tanner type with inverse trophic relation time delay, that is where the prey eaten by the mature predator can consume the immature predators. Let us mention that the first equation of system is stard. By contrast, the second equation is absolutely not stard. This intactness model contains a modified Leslie Gower term, the second term on the right-h side in the second equation of. The last depicts the loss in the predator population. The Leslie Gower formulation is based on the assumption that reduction in a predator population has a reciprocal relationship with per capita availability of its preferred food. Indeed, Leslie introduced a predator prey model where the carrying capacity of the predator environment is proportional to the number of prey. He stresses the fact that there are upper limits to the rates of increase of both prey x predator y, which are not recognized in the Lotka Volterra model. In case of continuous time, the considerations lead to the following: dy dt = a y y, αx in which the growth of the predator population is of logistic form, i.e. dy dt = a y y. C Here, C measures the carry capacity set by the environmental resources is proportional to prey abundance, C =αx, where α is the conversion factor of prey into predators. The term y/αx of this equation is called the Leslie Gower term. It measures the loss in the predator population due to the rarity per capita y/x of its favorite food. In the case of severe scarcity, y can switch over to other population, but its growth will be limited by the fact that its most favorite food, the prey x, is not available in abundance. The situation can be taken care of by adding a positive constant to the denominator, hence the equation above becomes, dy dt = a y y αx d thus, dy dt = y a a α. y x d α that is the second equation of system. In this paper, we introduce time delays in model, which is a more realistic approach to the understing of predator prey dynamics. Time delay plays an important role in many biological dynamical systems, being particularly relevant in ecology, where time delays have been recognized to contribute critically to the stable or unstable outcome of prey densities due to predation. Therefore, it is interesting important to study the following delayed modified Leslie Gower Holling-Type-II schemes: ẋt = a bxt c yt xt ẏt = a c yt r yt xt r xt, for all t>. Here, we incorporate a single discrete delay r> in the negative feedback of the predator s density.
6 A.F. Nindjin et al. / Nonlinear Analysis: Real World Applications 7 6 4 8 Let us denote by R the nonnegative quadrant by intr the positive quadrant. For θ [r, ], we use the following conventional notation: x t θ = xθ t. Then the initial conditions for this system take the form { x θ = θ, y θ = θ 3 for all θ [r, ], where, C[r, ], R, x = > y = >. It is well known that the question of global stability of the positive steady state in a predator prey system, with a single discrete delay in the predator equation without instantaneous negative feedback, remains a challenge, see [3,5,7]. Our main purpose is to present some results about the global stability analysis on a system with delay containing modified Leslie Gower Holling-Type-II terms. This paper is organized as follows. In the next section, we present some preliminary results on the boundedness of solutions for system 3. Next, we study some equilibria properties for this system give a permanence result. In Section 5, the analysis of the global stability is made for a boundary equilibrium sufficient conditions are provided for the positive equilibrium of both instantaneous system system with delay 3 to be globally asymptotically stable. Finally, a discussion which includes local stability results for system 3 is given.. Preliminaries In this section, we present some preliminary results on the boundedness of solutions for system 3. We consider x, y a noncontinuable solution, see [4], of system 3, defined on [r, A[, where A ], ]. Lemma. The positive quadrant intr is invariant for system. Proof. We have to show that for all t [,A[, xt> yt>. Suppose that is not true. Then, there exists <T <Asuch that for all t [,T[, xt> yt>, either xt = oryt =. For all t [,T[, we have xt = x exp a bxs c ys ds xs yt = y exp a c ys r ds. 5 xs r As x, y is defined continuous on [r, T ], there is a M such that for all t [r, T [, xt = x exp a bxs c ys xs ds x exptm yt = y exp a c ys r ds y exptm. xs r Taking the limit, as t T, we obtain xt x exptm> yt y exptm>, which contradicts the fact that either xt = oryt =. So, for all t [,A[, xt> yt>. 4
Lemma. For system 3, A = A.F. Nindjin et al. / Nonlinear Analysis: Real World Applications 7 6 4 8 7 lim sup xt K 6 lim sup yt L where K = a /b L = a /c K e ar. 7 Proof. From the first equation of system 3, we have for all t [,A[, ẋt < xta bxt. A stard comparison argument shows that for all t [,A[, xt xt where x is the solution of the following ordinary differential equation { xt = xta b xt, x = x>. As lim xt = a /b, then x thus x is bounded on [,A[. Moreover, from Eq. 5, we can define y on all interval [kr, k r], with k N, it is easy to see that y is bounded on [,A[ if A<. Then A =, see [4, Theorem.4]. Now, as for all t, xt xt, then lim sup xt lim sup xt = K. From the predator equation, we have ẏt<a yt, hence, for t>r, yt yt re a r, which is equivalent, for t>r,to yt r yte a r. 8 Moreover, for any μ >, there exists positive T μ, such that for t>t μ, xt<μk. According to 8, we have, for t>t μ r, ẏt<yt a c e a r yt, μk which implies by the same arguments use for x that, lim sup yt L μ, where L μ = a /c μk e ar. Conclusion of this lemma holds by letting μ. 3. Equilibria In this section we study some equilibria properties of system 3. These steady states are determined analytically by setting ẋ =ẏ =. They are independent of the delay r. It is easy to verify that this system has three trivial boundary equilibria, E =,, E = a /b, E,a /c.
8 A.F. Nindjin et al. / Nonlinear Analysis: Real World Applications 7 6 4 8 Proposition 3. System 3 has a unique interior equilibrium E =x,y i.e. x > y > if the following condition holds a < a. 9 c c Proof. From system 3, such a point satisfies a bx x = c y y = a x. c If 9 holds, this system has two solutions x,y x,y given by x ± = c b c a a c c b ± Δ /, y± = a x±, c where Δ = c a a c c b 4c bc a c a >. Moreover, it is easy to see that, x > x <. Linear analysis of system 3 shows that point E is unstable it repels in both x y directions point E is also unstable it attracts in the x-direction but repels in the y-direction. For r>, the characteristic equation of the linearized system at E takes the form where P λ Q λe λr =, P λ = λ Aλ, Q λ = a λ A, A = a c a. c Let us define F y = P iy Q iy. It is easy to verify that the equation F y = has one positive root. Therefore, if E is unstable for r =, it will remain so for all r>, if it is stable for r =, there is a positive constant r, such that for r>r, E becomes unstable. It is easy to verify that for r =, E is asymptotically stable if a /c >a /c, stable but not asymptotically if a /c = a /c unstable if a /c <a /c. 4. Permanence results Definition 4. System 3 is said to be permanent, see [4], if there exist α, β, < α < β, independent of the initial condition, such that for all solutions of this system, { min lim inf xt,lim inf } yt α
A.F. Nindjin et al. / Nonlinear Analysis: Real World Applications 7 6 4 8 9 { max lim sup xt,lim sup } yt β. Theorem 5. System 3 is permanent if L< a c. Proof. By Lemma, there is a β = max{k,l} > independent of the initial condition such that { } max lim sup xt, lim sup yt β. We only need to show that there is a α >, independent of the initial data, such that { } min lim inf xt,lim inf yt α. It is easy to see that, for system 3, for any μ > for t large enough, we have yt<μl. Thus, we obtain ẋ>x a bx c μl. By stard comparison arguments, it follows that lim inf xt a c μl b letting μ, we obtain lim inf xt b a c L. 3 Let us denote by N = /ba c L/. If is satisfied, N >. From 3 Lemma for any μ >, there exists a positive constant, T μ, such that for t>t μ, xt>n /μ yt<μl. Then, for t>t μ r, wehave ẏt>yt a μc yt r. 4 N μ On the one h, for t>t μ r, these inequalities lead to ẏt> μ c L yt, N μ which involves, for t>t μ r, μ c L yt r<ytexp r. 5 N μ On the other h, from 4 5, we have for t>t μ r, ẏt>yt a μc N μ exp μ c L N μ r yt which yields lim inf yt a N μ exp μ c L r = y μ. μc N μ
A.F. Nindjin et al. / Nonlinear Analysis: Real World Applications 7 6 4 8 Letting μ, we get lim inf yt a N exp c L r = y. c N Let be α = min{n,y } >. Then we have shown that system 3 is permanent. 5. Global stability analysis 5.. Stability of E Theorem 6. If a c y <, K then E is globally asymptotically stable for system 3. Proof. As lim inf yt y lim sup xt K, from the prey s equation we obtain: μ, T μ >, t >T μ, As ẋt<xt a bx a c y <, K there exists μ > such that a c y μkμ <. c y μkμ. Then, by stard comparison arguments, it follows that lim sup xt thus, lim xt =. The ω-limit set Ω of every solution with positive initial conditions is then contained in {, y, y }. Now from 7, we obviously obtain Ω {,y, y L}. As E / Ω, E is unstable is repels in both x y directions as Ω is nonempty closed invariant set, therefore Ω ={E }. 5.. Stability of E without delay First, we give some sufficient conditions which insure that the steady state in the instantaneous system, i.e. without time delay, is globally asymptotically stable. Theorem 7. The interior equilibrium E is globally asymptotically stable if a c <b x, 6 a a <b c a. 7 Proof. The proof is based on constructing a suitable Lyapunov function. We define x y Vx,y = x x x ln x α y y y ln y, where α = c / a.
A.F. Nindjin et al. / Nonlinear Analysis: Real World Applications 7 6 4 8 This function is defined continuous on intr. It is obvious that the function V is zero at the equilibrium E is positive for all other values of x y, thus, E is the global minimum of V. The time derivative of V along a solution of system 3 is given by dv = x x a bx c y αy y a c y. dt x x Centering dv/dt on the positive equilibrium, we get dv = b a bx x x c α y y dt x x k a α c x x y y x x = b a bx x x c α y y 8 x x a α c x x y y. x x b a bx c x x k c α c y y x b a bx c x x c x x c a From 6, we obtain b a bx c <. From 6 7, there exists μ > T>, such that μk c <, a for t>t, dv b dt a bx c c x x x μk c a y y. 9 y y. Thus, dv/dt is negative definite provided that 6 7 holds true. Finally, E is globally asymptotically stable. If in particular, we suppose that environment provides the same protection to both prey predator i.e. = then Theorem 7 can be simplified as follows. Corollary 8. The interior equilibrium E is globally asymptotically stable if = a <b x.
A.F. Nindjin et al. / Nonlinear Analysis: Real World Applications 7 6 4 8 Proof. From 8, we have dv = b a bx dt x k c x x x b a bx k c x x x x x c α x y y x x y y x x c α x y y x x y y, which is negative definite provided that holds true thus E is globally asymptotically stable. 5.3. Stability of E with delay In this subsection we shall give a result on the global asymptotic stability of the positive equilibrium for the delayed system. Theorem 9. Assume that parameters of system 3 satisfy a c > max { a c, 3 a c }. Then, for b large enough, there exists r > such that, for r [,r ], the interior equilibrium E is globally asymptotically stable in R. Proof. First of all, we rewrite Eqs. to center it on its positive equilibrium. By using the following change of variables, xt Xt = ln x yt Yt= ln, y the system becomes Ẋt = x b a bx e Xt c y e Yt, xt xt c y Ẏt= e Y a x 3 e X. xt r xt r According to the global existence of solutions established in Lemma, we can assume that the initial data exists on [r, ] this can be done by changing initial time. Now, let μ > be fixed let us define the following Liapunov functional V : C[r; ], R R, φ φ Vφ, φ = e u du e u du a x e φ u du r c y e φ s c y e φ sr a x e φ sr ds dv r v c c y rμl e φ s a x e φ s ds. r Let us define the continuous nondecreasing function u : R R by ux = e x x.
We have u =, ux > for x> A.F. Nindjin et al. / Nonlinear Analysis: Real World Applications 7 6 4 8 3 u φ uφ uφ Vφ, φ, where φ = φ, φ denotes the infinity norm in R. Let v : R R be define by vx = ux e x a x r c y a x c y r e x c r μl. 4 It is clear that v is a continuous nondecreasing function satisfies v =, vx > Vφ, φ v φ, where denotes the infinity norm in C[r; ], R.Now,letX, Y be a solution of 3 let us compute V 3, the time derivative of V along the solutions of 3. First, we start with the function φ φ V φ, φ = e u du e u du a x e φ u du. r We have Xt Yt V X t,y t = e u du e u du a x e Xu du. Then, V 3 = e Xt Ẋt e Yt Ẏt a x e Xt e X. System 3 gives us V 3 = x b a bx e Xt c y e Yt e Xt xt xt c y e Yt e Y xt r a x e Yt e X xt r a x e Xt e X. By using several times the obvious inequalities of type c y e Yt e Xt c y xt xt e Yt ext c y e Yt e Xt, we get V 3 x b a bx a c y e Xt k c y a x e Yt c y e Yt e Y. xt r
4 A.F. Nindjin et al. / Nonlinear Analysis: Real World Applications 7 6 4 8 As, e Y = e Yt e Ys Ẏsds, we obtain V 3 x b a bx a c y k c y a x c y xt r c y e Yt xt r From 3, we have e Yt e Ys Ẏsds e Ys e Ys e Xt e Yt e Ys Ẏsds. 4 c y = e Yt e Ysr ds xs r a x e Yt e Xsr ds xs r k c y a x e Ys ds.e Yt c y e Ys e Ysr ds a x e Ys e Xsr ds as y e Ys μl for s large enough, we obtain, for t large enough, e Yt e Ys c Ẏsds μlr a x y e Ys c y e Ysr a x e Yt e Xsr as xs r μl for s large enough, 4 becomes, for t large enough, V 3 x b a bx a c y e Xt k c y a x c y c μlr c y a x e Yt xt r c y e Ys c y e Ysr a x e Xsr ds. 5 The next step consists in computation of the time derivative along the solution of 3, of the term V φ, φ = c y r v c rμl ds e φ s c y e φ sr a x r c y e φ s a x e φ sr e φ s ds. ds dv
We have Then, V X t,y t = c y A.F. Nindjin et al. / Nonlinear Analysis: Real World Applications 7 6 4 8 5 c rμl v V 3 = c y e Yt c y e Xsr e Ys c y e Ysr a x k c y e Ys a x e Xs e Y a x e X r ds. c y e Ys c y e Ysr a x e Xsr c c y rμl e Yt a x e Xt c c y rμl e Y a x e X = c r c y e Y a x e X y e Yt μl c y e Ys c y e Ysr a x e Xsr c c y rμl e Yt a x e Xt. ds ds ds dv For t large enough, we have y e Yt μl< thus, V 3 c y e Ys c y e Ysr a x e Xsr ds c c y rμl e Yt a x e Xt. This inequality 5 lead to, for t large enough, V 3 x b a bx a c y e Xt k c y a x c y c μlr c y a x e Yt xt r c c y rμl e Yt a x e Xt k bx x a x a x c y c a x rμl e Xt Now, if bx x c y c y a x a x a x a x c y xt r c μlr c y a x e Yt. c y < 6 c y xt r < 7
6 A.F. Nindjin et al. / Nonlinear Analysis: Real World Applications 7 6 4 8 then, for r small enough, we can conclude that V 3 is negative definite. Hence V satisfies all the assumptions of Corollary 5. in [4] the theorem follows. We now study when inequalities 6 7 hold. From, we have bx x = c y a x /c using, 6 becomes 3 c a x c which is rewritten as x a c 3a c a a c 3a c a a x < As x a b, then the following inequality <b < a c 3 a c. 8 a c 3 a c implies 8. Now, if the following inequality holds c y a x c y a <, 3 b then, for t large enough, 7 holds too. Using, 3 is reformulated as c y c y a < c y a b c c < c a a b y c c < c a b c x c <c a b k x. k As x a /b, then the last inequality is satisfied if c <c 3 a b k k that is if a c a c <b c c. 3 In conclusion, if { a a > max, 3 } a c c c for b large enough, there exists a unique interior equilibrium for r small enough it is globally asymptotically stable. 9 3
A.F. Nindjin et al. / Nonlinear Analysis: Real World Applications 7 6 4 8 7 6. Discussion It is interesting to discuss the effect of time delay r on the stability of the positive equilibrium of system 3. We assume that positive equilibrium E exists for this system. Linearizing system 3 at E, we obtain Ẋt = A Xt A Yt, Ẏt= A Xt r A Yt r, where A =bx c x y x, A = c x x, A = c y x = a c A = c y x =a. The characteristic equation for 33 takes the form in which Pλ Qλe λr = 34 Pλ = λ A λ Qλ =A λ A A A A. When r =, we observe that the jacobian matrix of the linearized system is A A J =. A A One can verify that the positive equilibrium E is stable for r = provided that a <b. 35 Indeed, when 35 holds, then it is easy to verify that By denoting we have T rj = A A < DetJ = A A A A >. Fy= P iy Qiy, Fy= y 4 A A y A A A A. If 35 holds, then λ = is not a solution of 34. It is also easy to verify that if 35 holds, then equation Fy= has at least one positive root. By applying stard theorem on the zeros of transcendental equation, see [4, Theorem 33
8 A.F. Nindjin et al. / Nonlinear Analysis: Real World Applications 7 6 4 8 4.], we see that there is a positive constant r which can be evaluated explicitly, such that for r>r, E becomes unstable. Then, the global stability of E involves restrictions on length of time delay r. Therefore, it is obvious that time delay has a destabilized effect on the positive equilibrium of system 3. References [] M.A. Aziz-Alaoui, Study of Leslie Gower-type tritrophic population, Chaos Solitons Fractals 4 8 75 93. [] M.A. Aziz-Alaoui, M. Daher Okiye, Boundeness global stability for a predator prey model with modified Leslie Gower Holling-typeII schemes, Appl. Math. Lett. 6 3 69 75. [3] E. Beretta, Y. Kuang, Global analyses in some delayed ratio-depended predator prey systems, Nonlinear Anal. Theory Methods Appl. 3 3 998 38 48. [4] Y. Kuang, Delay Differential Equations, with Applications in Population Dynamics, Academic Press, New York, 993. [5] R.K. Upadhyay, V. Rai, Crisis-limited chaotic dynamics in ecological systems, Chaos Solitons Fractals 5 8. [6] R.K. Upadhyay, S.R.K. Iyengar, Effect of seasonality on the dynamics of 3 species prey predator system, Nonlinear Anal.: Real World Appl. 6 5 59 53. [7] R. Xu, M.A.J. Chaplain, Persistence global stability in a delayed predator prey system with Michaelis Menten type functional response, Appl. Math. Comput. 3 44 455.