Nonlinear Analysis: Real World Applications. Global dynamics of Nicholson-type delay systems with applications

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1 Nonlinear Analysis: Real World Applications (0) Contents lists available at ScienceDirect Nonlinear Analysis: Real World Applications journal homepage: Global dynamics of Nicholson-type delay systems with applications L. Berezansky a, L. Idels b,, L. Troib a a Department of Mathematics, Ben-Gurion University of Negev, Beer-Sheva 8405, Israel b Department of Mathematics, Vancouver Island University, 900 Fifth St., Nanaimo, BC, Canada V9S5S5 a r t i c l e i n f o a b s t r a c t Article history: Received 7 January 00 Accepted 6 June 00 Keywords: Nicholson-type delay differential systems Global and local stability Equilibria Permanence Population dynamics Marine protected area (MPA) B-cell chronic lymphocytic leukemia (B-CLL) dynamics Models of marine protected areas and B-cell chronic lymphocytic leukemia dynamics that belong to the Nicholson-type delay differential systems are proposed. To study the global stability of the Nicholson-type models we construct an exponentially stable linear system such that its solution is a solution of the nonlinear model. Explicit conditions of the existence of positive global solutions, lower and upper estimations of solutions, and the existence and uniqueness of a positive equilibrium were obtained. New results, obtained for the global stability and instability of equilibria solutions, extend known results for the scalar Nicholson models. The conditions for the stability test are quite practical, and the methods developed are applicable to the modeling of a broad spectrum of biological processes. To illustrate our finding, we study the dynamics of the fish populations in Marine Protected Areas. 00 Elsevier Ltd. All rights reserved.. Introduction We consider a metapopulation of some species that grow by local dynamics and interact in patches coupled by migration (dispersion). A general model, which is capable of depicting a broad spectrum of biological features [,], can be expressed as dx = G(x(t τ)) Mx(t) + Dx(t), where the vector function x(t) is a population size that maps [0, ) into R n ; the nonlinear function G : R n R n is the birth rate; the n n diagonal matrix M represents the mortality rate; the n n matrix D is the dispersal between patches, and τ > 0 is the maturation period. System () occurs in a number of applications. ().. Marine protected areas model Marine protected areas and marine reserve areas have been promoted as conservation and fishery management tools to hedge marine life and sustain ecosystems [3]. To describe the ecological linkage between the reserve and the fishing ground, we will consider two regions A and A. Let t denotes a time and a chronological age. We define the following functions: u = u (t, a) is the age distribution of the fish population in the protected area (reserve) A ; u = u (t, a) is the age distribution of the fish population in the fishing area A ; m (t, a) is the natural mortality rate in A ; m (t, a) is the Corresponding author. Tel.: ; fax: addresses: brznsky@cs.bgu.ac.il (L. Berezansky), lev.idels@viu.ca (L. Idels), ltroib@gmail.com (L. Troib) /$ see front matter 00 Elsevier Ltd. All rights reserved. doi:0.06/j.nonrwa

2 L. Berezansky et al. / Nonlinear Analysis: Real World Applications (0) natural mortality rate in A. Let d be a net transfer rate, i.e., some net flow of adult fishes from the reserve, and let d be the immigration rate from the fishing area to the reserve, say larval dispersion; h(t, a) is the harvesting rate. To model the age structure of the population we will use [] a linear Foerster McKendrick system: t + u (t, a) = (m (t, a) + d ) u (t, a) + d u (t, a) a t + () u (t, a) = (m (t, a) + d ) u (t, a) + d u (t, a) h(t, a)u (t, a) a with u i (0, a) = ω i (a) 0 (i =, ), where ω i (a) denotes the initial conditions. If τ 0 is the maturation time, then the total matured population x i (t) at time t is defined as where x i (t) = u i (t, a)da, τ u i (t, 0) = γ i x i e α ix i is the birth function with positive constants α i and γ i (i =, ). It is biologically reasonable to assume that only mature fish (with a > τ ) can reproduce, and the reproduction rate depends on the mature population. Integration along characteristics of system () yields the resulting model the matured population dx = γ x (t τ)e α x (t τ) m x (t) + d x (t) d x (t) dx = γ x (t τ)e α x (t τ) m x (t) + d x (t) d x (t) hx (t), and x (t) = ϕ (t) 0, x (t) = ϕ (t) 0 for t [ τ, 0]. (3).. Novel two-compartment model of cancer cell population The canonical model of cancer cell population dynamics consists of proliferating and quiescent cell compartments; cell populations grow by a one-hump curve, and this model allows for transitions between the compartments [4]. A compartmental model of B-cell chronic lymphocytic leukemia (B-CLL) dynamics can be expressed as db = b(t)g (b(t)) m b(t) + d l(t) d b(t) dl = l(t)g (l(t)) m l(t) + d b(t) d l(t), where b(t) is the number of normal cells; l(t) is the number of leukemic cells; G is a per capita growth rate; m is the death rate of the normal cells (a certain proportion of the normal cells get destroyed due to inhibition from the leukemic cells and also to natural causes of death). In the second equation, G is the per capita growth rate of the leukemic cells (L-cells); m is the death rate of the leukemic cells (a certain proportion of leukemic cells become inactive as a result of self-inhibition). Note that m m ; thus the effect of m is too small to have a significant impact on the dynamics of L-cells. The constants d i in both equations represent the dispersal (transition) rates. Classical models have been the mainstay for models of cell growth (see for example [5] or [6]), based on the assumption that the mechanism of the growth rate of both cells is a Gompertzian curve xg(x) = x cx ln max x. Function G(x) is called an inhibition function. The key assumption embodied in the Gompertz model is that the cell growth rate decreases exponentially as a function of time. Note also that for the Gompertz model the inhibition logarithmic function G(x) and its derivative are more likely to cause chaotic (abruptive) behavior in relatively slower growing populations. According to the recent experimental data [7,8] (see also [4]), the Gompertz model is not suitable for extrapolating the specific growth rate (or generation time) of the cells when the concentration is low and/or at the early stage of cell development. We assume that the cell growth rate decreases exponentially as a function of population size, rather than a function of time, e.g. we choose a hump-shaped skewed to the right smooth function xg(x) = rxe kx. To present the cell growth as exponentially decreasing functions of the cell populations, we set up in system (4) the inhibition functions G (b) = γ e αb(t) and G (l) = γ e βl(t). In any cell growth, some cells are inactive and, once activated, the cell division is not instantaneous. The inclusion of explicit time lags in the model allows direct reference to experimentally measurable and/or controllable cell growth (4)

3 438 L. Berezansky et al. / Nonlinear Analysis: Real World Applications (0) characteristics: e.g. the time required to perform the necessary divisions. Let τ be the time required for the L-cells to respond to growth signals to re-enter the cell cycle. We also assume no programmed cell death in the L-cells, whereas the B-cells undergo rapid cell death. Finally, we consider a B-CLL model expressed as the system db = γ b(t)e αb(t) m b(t) + d l(t) d b(t) dl = γ l(t τ)e βl(t τ) + d b(t) d l(t). Inspired by these two models (3) and (5), we consider the following nonlinear system: dx = a x (t) + b x (t) + c x (t τ) exp( x (t τ)) dx = a x (t) + b x (t) + c x (t τ) exp( x (t τ)), where a i, b i, c i, τ 0, with initial conditions x i (0) 0, x (t) = ϕ (t) 0, x (t) = ϕ (t) 0 (7) for t [ τ, 0]. Then system (6) has the following vector form dx = AX + F(X(t τ)), where [ ] a b A =, b a [ ] x (t) X(t) =, F(X(t τ)) = x (t) with f i (u) = c i u exp( u), u 0, (i =, ). [ ] f (x (t τ)) f (x (t τ)) Remark.. Without migration (b = b = 0), the delay differential model (6) is a direct extension of the well-known Nicholson model [9] (5) (6) (8) dx dx = a x + c x (t τ) exp( x (t τ)) = a x + c x (t τ) exp( x (t τ)). (9) We call model (6) a Nicholson-type delay differential system. The scalar Nicholson model is well studied (see the recent review [9] and references therein), whereas the Nicholson systems have not yet been studied. The qualitative theory of various delay differential systems was studied in monographs [0,,] and the most recent papers [ ]. Numerous applications of delay differential systems can be found in [9,3 8]. In a recent paper [9], the authors studied the global dynamics of a class of quasi-linear system dx m = Ax(t) + f k (x(t τ k )), k= where A is a diagonal matrix, and f k is a Lipschitzian function. The explicit conditions of existence and uniqueness of an equilibrium and global stability of system (0) were obtained, and applied to delayed cellular neural network, (BAM) neural networks and some population growth models. Remark.. Note that for the Nicholson-type delay models (8) these results and techniques are not applicable since in model (6) matrix A is not a diagonal matrix, and for this model the function F is not a Lipschitzian function. The subject of this paper is the study of global dynamics of a new nonlinear system of delay differential equations, and application of these results to novel population biology models (3) and (5), that are united by the common theme of possessing a regime of dynamic behavior with migration and control. The paper is organized as follows. In Section, the study of positive solutions, boundedness of solutions, and lower and upper estimations of solutions is presented. In Section 3, we prove the existence and uniqueness of a positive equilibrium. We study the global stability and instability of the zero solution in Section 4, whereas, in Section 5, we study the local and global dynamics of the positive equilibrium. Finally, in Section 6, we apply our results to establish threshold criteria for extinction and permanence of the fish population in marine protected areas. (0)

4 L. Berezansky et al. / Nonlinear Analysis: Real World Applications (0) Positiveness, boundedness and permanence of global solutions Theorem.. There is a unique global positive solution of problem (6) (7) provided that x i (0) > 0, i =,. Proof. It follows from the standard existence theorem (see, for example [30]) that there exists a unique local solution of this problem. We have x (t) = x (0)e a t + t 0 e a (t s) [b x (s) + c x (t τ) exp( x (s τ))]ds > 0. Similarly, x (t) > 0. Thus we shall only prove that there exists a global solution. Suppose that y(t) = x (t) + x (t) exists only on the interval [0, t 0 ) and lim t t0 y(t) = +. Since then max xe x = x 0 e, ẏ(t) max{b, b }y(t) + c e + c e. Hence 0 < y(t) z(t), where z(t) is a solution of the problem ż(t) = max{b, b }z(t) + c e + c e, z(0) = x (0) + x (0). The solution of this equation is bounded on any finite interval; thus lim t t0 z(t) < +. That contradiction proves that problem (6) (7) has a global solution. Theorem.. Suppose that = a a b b > 0. Then the solution X(t) = {x (t), x (t)} T of (6) (7) is bounded; moreover, and lim sup x (t) t e [a c + b c ], lim sup x (t) t e [b c + a c ]. () () Proof. To get started, we notice that ẋ (t) a x (t) + b x (t) + c ẋ (t) a x (t) + b x (t) + c e. Hence, for any t > t 0, x (t) e a (t t 0 ) x (t 0 ) + b t x (t) e a (t t 0 ) x (t 0 ) + b t where d = c ea, d = c ea. Then Hence e t 0 e a (t s) x (s)ds + d t 0 e a (t s) x (s)ds + d, x (t) e a (t t 0 ) x (t 0 ) + b sup x (s) + d. a t 0 s lim sup x (t) b sup x (s) + d. t a t 0 s

5 440 L. Berezansky et al. / Nonlinear Analysis: Real World Applications (0) Since t 0 > 0 is an arbitrary point, then x b a x + d, where x = lim sup t x (t), x = lim sup t x (t). Similarly, If x b a x + d. b B = a b, a ] [ ] [ x d X =, D = x d then B X D. Clearly, the inequality B 0 yields X B D, which proves () (). Theorem.3. Suppose that c > a > 0 and c > a > 0. Then the solution of system (6) (7) is bounded from below by a positive constant; moreover, lim inf x (t) c e c a e t ea lim inf x (t) c e c a e. t ea (3) Proof. From the first equation of system (6), we have ẋ (t) a x (t) + c x (t τ) exp( x (t τ)). (4) Consider the following scalar equation: ẏ(t) = ay(t) + cy(t τ) exp( y(t τ)) with positive initial conditions. If c > a > 0, then, by Theorem 3.6 of [9], lim inf t y(t) c ea e c ae. The proof of Theorem 3.6 in [9] is also valid for solutions of the differential inequality (4). Therefore, lim inf x (t) c e c a e. t ea The second inequality, (3), can be proven similarly. Definition. System (6) is permanent if there exist two numbers m and M (0 < m M) such that, for any solution of system (6), we have m lim inf t x i(t) lim sup x i (t) M, i =,. t Based on Theorems. and.3, we obtain the following result. Theorem.4. System (6) is permanent if the following conditions hold: > 0, c > a > 0 and c > a > Existence of positive internal equilibrium Theorem 3.. () If b > 0, b > 0, c > a > 0, c > a > 0, and > 0, then there exists a unique positive equilibrium of Eq. (6).

6 L. Berezansky et al. / Nonlinear Analysis: Real World Applications (0) () If there exists a unique positive equilibrium of Eq. (6), then c > a b or c > a b. Proof.. To prove that system (6) has a unique internal equilibrium, we consider the following system: a x + b x + c x e x = 0 L a x + b x + c x e x = 0 L. Curve L intersects the x -axis at x = ln c a, whereas curve L intersects the x -axis at x = ln c a. For curve L, x = x b (a c e x ) := f (x ). Clearly, x > 0 if x < x, and x < 0 for x > x. Similarly, for curve L, x = x b (a c e x ) := f (x ), and x > 0 if x < x, and x < 0 for x > x. Therefore, it is sufficient to consider the behavior of both curves only for x > x and x > x. For curve L, x = b [a c ( x )e x ]. If x then x > 0; if 0 < x < then inequality x > ln c implies that a x > b [a c e x ] > 0. Hence x = f (x ) is an increasing function. Similarly, f (x ) > 0; hence there exists an inverse function x = f (x ), and function f is an increasing function. We have f (x ) > f (x ) = 0. Curve L has the asymptote x = a x b, and L has the asymptote x = b x a. The assumption > 0 guarantees that a b > b ; a thus the first line is above the second one. Hence, for sufficiently large x, we have f (x ) > f (x ); therefore curves L and L have a positive point of intersection. To prove that this point is a unique equilibrium, we describe curves L and L in polar coordinates, where θ will be the polar angle. For curve L, the function a c e x θ = arctan = φ(x ) b is an increasing function for x > x. For curve L, function b θ = arctan = ψ(x a c e x ) is a decreasing function for x > x ; on the other hand, x = f (x ) for this curve is an increasing function of x ; thus for curve L a polar angle is a decreasing function. If a positive point of intersection of curves L and L exists, then the opposite monotonicity of functions φ and ψ guarantees that the positive point of intersection is unique. To prove part of Theorem 3., we assume that (x, x ) is a positive point of intersection of the curves L and L. Then x x = (a c e x x ), and b x = (a c e x ). b Hence (a c e x ) or (a c e x ). b b Finally, c (a b )e x > a b or c (a b )e x > a b.

7 44 L. Berezansky et al. / Nonlinear Analysis: Real World Applications (0) Stability of the trivial solution Consider a nonlinear system with delay, dx = Ax(t) + F(t, x(t τ)), t > 0, τ > 0, x(t) = φ(t), t [ τ, 0], (5) where x : [ τ, ] R n, A R n n. Here F : R + R n R n is a nonlinear and continuous vector function. We assume that (5) has a unique global solution for t τ. Definition 4.. Suppose that system (5) has a trivial solution x = 0. This trivial solution is globally asymptotically stable if, for any solution of system (5), we have lim t x(t) = 0. Lemma 4. ([4]). Suppose that F(t, x) λ x, λ < µ(a), where is a norm in R n and µ(a) is the matrix measure I + ϵa µ(a) = lim. ϵ 0 + ϵ Then the trivial equilibrium of system (5) is globally asymptotically stable. Remark 4.. Function µ(a) is called a logarithmic matrix norm (see for example, [3]). Theorem 4.. Suppose that max{c, c } < min{a b, a b }. Then the trivial solution of system (6) is globally asymptotically stable. Proof. Consider system (6) with initial conditions (7) in a vector form (8). There exists v(0 < v < u) such that f (u) = f (v)u, u 0, where f (v) = c e v ( v). We have Hence f (0) = c, f (v) = c e v ( v). sup f (v) = max{f (0), f () } = max{c, c e } = c. u 0 Then f (u) c u. Similarly, f (u) c u. If we choose the norm X = max{x, x } in the space R, then F(X) max{c, c } X. For the chosen norm in R, we have µ(a) = max{ a +b, a +b } = min{a b, a b }. Lemma 4. finalizes the proof of the theorem. Theorem.3 implies the following result. Theorem 4.. Suppose that c > a and c > a. Then the trivial solution of system (6) is not asymptotically stable. Proof. By Theorem.3, any solution of system (6) is bounded from below by a positive constant; therefore lim t x i (t) Local and global stability of positive equilibrium In the following discussion, we say that a positive solution of system (6) is globally asymptotically stable if it attracts all other positive solutions of the system (see, for example []). We begin with the local stability conditions. Note that system (6) with dispersion has two parts: nonlinear terms and linear perturbation; and the standard procedure of linearization turns the system into a linear system with not only positive coefficients. Theorem 5.. Suppose that a positive internal equilibrium (x, x ) of system (6) exists and max{c x e x, c x e x } < min{a b, a b }. Then this equilibrium is a local attractor.

8 L. Berezansky et al. / Nonlinear Analysis: Real World Applications (0) Proof. The substitutions y i = y i (t) = x i (t) x i and y iτ = y i (t τ) = x i (t τ) x i into system (6) yield dy = a y + b y a x + b x + c (y τ + x )e (y τ +x ) dy = a y + b y a x + b x + c (y τ + x )e (y τ +x ). By the standard linearization procedure for nonlinear system (6), we obtain the following linear system: (6) dy dy = a y + b y + c e x ( x )(y τ ) = a y + b y + c e x ( x )(y τ ). Lemma 4. implies that system (7) is asymptotically stable; hence the positive equilibrium of nonlinear system (6) is locally asymptotically stable. Theorem 5.. Suppose that a positive internal equilibrium (x, x ) of system (6) exists, and that for some ϵ > 0 max{c e, c e, c e x ϵ, c e x ϵ} < min{a b, a b }. Then this equilibrium is a global attractor. Proof. Consider a special case ϵ = 0. Let f (u) = a x + b x + c (u + x )e (u+x ) ; then f (0) = 0, and Clearly, f (u) = c ( u x )e (u+x ), f (u) = c ( u x )e (u+x ). f (u) = f (c) u, where 0 < c < u for u 0, and for u < 0 we have x < c < 0.. Let x > 0, then for u 0 we have Hence max f (u) = max f ( x ), f (0), f ( x ) = c e, and 0 < f (0) < c e x. f (u) max{c e, c e x }. Note that f (u) is a convex function for x < u < 0; hence, for this interval, Since then f (u) < a x b x x u. a x + b x + c x e x = 0, a x b x = c e x ; x hence, for x < u < 0, f (u) < c e x u. By summing up, for the case x > 0 we proved that (7) f (u) < max{c e, c e x } u for x < u <. (8)

9 444 L. Berezansky et al. / Nonlinear Analysis: Real World Applications (0) Consider the case x < 0. Let x 0 be the negative root of the equation f (u) = 0. Then, for u x 0, we have f (u) f ( x ) = c e. If x < u < x 0, then f (u) < a x b x x u = c e x u. Therefore, for the case x < 0, the same inequality (8) holds. Lemma 4. completes the proof of the theorem for ϵ = 0. Let ϵ > 0. By Theorem.3, there exists δ > 0 such that, for any solution of system (6), we have y i (t) x +δ, i =,. We can improve inequality (8). There exists ϵ > 0 such that f (u) < max{c e, c e x ϵ} u (9) for δ x < u <. We will prove Theorem 5. if we repeat the steps taken for the proof of the case ϵ = 0, with inequality (9). Remark 5.. Note that, if b = b = 0, system (6) consists of two decoupled Nicholson scalar equations (9), and the results obtained in Theorems 5. and 5. coincide with the well-known stability condition a < c < ae. Example 5.. Consider the following symmetric system: dx = ax (t) + bx (t) + cx (t τ) exp( x (t τ)) dx = ax (t) + bx (t) + cx (t τ) exp( x (t τ)). Simple calculations prove that for c > a b > 0 there exists a unique equilibrium x = x = ln c of system (0). a b Theorems 5. and 5. yield the stability condition: c < (a b)e. This condition for the decoupled system (b = 0) coincides with the known stability condition c < ae for the scalar Nicholson equation (see for example [9]). (0) 6. Applications to marine protected area models In system (6), let c i = γ i > 0, a = m + d > 0, a = m + d + h > 0, b = d, and b = d ; then the system takes the form of (3). Thus, based on Theorem 3., we have the following result. Theorem 6.. If γ > m + d and γ > m + d + h then there exists a unique positive equilibrium of Eq. (3). Note that the condition > 0 of Theorem 3. is true for any choice of all parameters = a a b b = (m + d )(m + d + h) d d > 0. We would like to point out that the assumptions γ > m + d and γ > m + d + h are biologically motivated. To avoid population extinction under excessive harvesting, the loss in every region should be less than the gain in the region. Thus, based on the assumptions of Theorem 6., we may conclude from Theorem 4. that if h γ (m + d ), i.e. the harvesting rate is higher than the difference gain minus loss, then the fish stock may collapse. We define H critical = γ (m + d ) as a threshold level that prevents overfishing by controlling the rate of harvest. From the point of view of fishery managers, the existence of globally attractive solutions obtained in Theorem 5. is necessary for planning harvesting strategies and sustaining the fishing grounds. Acknowledgements We wish to express thanks to Dr. A. Gibson (Biology Department at Vancouver Island University) whose comments significantly improved the design of the B-CLL model. The authors also would like to extend their appreciation to the anonymous referee for his helpful suggestions, which have greatly improved this paper. The first author s research was supported in part by the Israeli Ministry of Absorption. The second author s research was supported by a grant from VIU, Canada.

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