Global dynamics of Nicholson-type delay systems with applications
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 [1], [2], can be expressed as where the vector function is a population size that maps into ; the nonlinear function is the birth rate; the diagonal matrix represents the mortality rate; the matrix is the dispersal between patches, and is the maturation period.
System (1) occurs in a number of applications.
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 and . Let denotes a time and chronological age. We define the following functions: is the age distribution of the fish population in the protected area (reserve) ; is the age distribution of the fish population in the fishing area ; is the natural mortality rate in ; is the natural mortality rate in . Let be a net transfer rate, i.e., some net flow of adult fishes from the reserve, and let be the immigration rate from the fishing area to the reserve, say larval dispersion; is the harvesting rate.
To model the age structure of the population we will use [1] a linear Foerster–McKendrick system: with , where denotes the initial conditions. If is the maturation time, then the total matured population at time is defined as where is the birth function with positive constants and .
It is biologically reasonable to assume that only mature fish (with ) can reproduce, and the reproduction rate depends on the mature population. Integration along characteristics of system (2) yields the resulting model the matured population and for .
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 where is the number of normal cells; is the number of leukemic cells; is a per capita growth rate; 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, is the per capita growth rate of the leukemic cells (L-cells); is the death rate of the leukemic cells (a certain proportion of leukemic cells become inactive as a result of self-inhibition). Note that ; thus the effect of is too small to have a significant impact on the dynamics of L-cells. The constants 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 . Function 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 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 . To present the cell growth as exponentially decreasing functions of the cell populations, we set up in system (4) the inhibition functions and .
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 characteristics: e.g. the time required to perform the necessary divisions. Let 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
Inspired by these two models (3), (5), we consider the following nonlinear system: where , with initial conditions , for . Then system (6) has the following vector form where with Remark 1.1 Without migration , the delay differential model (6) is a direct extension of the well-known Nicholson model [9]
The qualitative theory of various delay differential systems was studied in monographs [10], [11], [2] and the most recent papers [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. Numerous applications of delay differential systems can be found in [9], [23], [24], [25], [26], [27], [28].
In a recent paper [29], the authors studied the global dynamics of a class of quasi-linear system where is a diagonal matrix, and is a Lipschitzian function. The explicit conditions of existence and uniqueness of an equilibrium and global stability of system (10) were obtained, and applied to delayed cellular neural network, (BAM) neural networks and some population growth models. Remark 1.2 Note that for the Nicholson-type delay models (8) these results and techniques are not applicable since in model (6) matrix is not a diagonal matrix, and for this model the function is not a Lipschitzian function.
Section snippets
Positiveness, boundedness and permanence of global solutions
Theorem 2.1 There is a unique global positive solution of problem (6)–(7) provided that .
Proof It follows from the standard existence theorem (see, for example [30]) that there exists a unique local solution of this problem. We have Similarly, . Thus we shall only prove that there exists a global solution. Suppose that exists only on the interval and . Since then
Existence of positive internal equilibrium
Theorem 3.1 Ifthen there exists a unique positive equilibrium of Eq.(6). If there exists a unique positive equilibrium of Eq.(6), then
Proof 1. To prove that system (6) has a unique internal equilibrium, we consider the following system: Curve intersects the -axis at , whereas curve intersects the -axis at . For curve , Clearly, if , and for
Stability of the trivial solution
Consider a nonlinear system with delay, where . Here is a nonlinear and continuous vector function. We assume that (15) has a unique global solution for .
Definition 4.1 Suppose that system (15) has a trivial solution . This trivial solution is globally asymptotically stable if, for any solution of system (15), we have .
Lemma 4.1 Suppose thatwhere is a norm in and is the matrix measure[24]
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 [11]). 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 5.1 Suppose that a positive internal equilibrium
Applications to marine protected area models
In system (6), let , , and ; then the system takes the form of (3). Thus, based on Theorem 3.1, we have the following result. Theorem 6.1 If and then there exists a unique positive equilibrium of Eq. (3).
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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