Properties of Solutions of a Class of Equations With Lag Which Describe the Dynamics of Change in the Population of a Species With the Age Structure Taken Into Account

Kolesov, Yu. S.

doi:10.1070/sm1983v045n01abeh002588

Cited by 14 publications

(9 citation statements)

References 5 publications

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“…The hypothesis of our theorem guarantees that such an equilibrium exists. Koslesov [13] and Mazorov [15] point out that equation (1) can possess periodic solutions, while Beddington and May [2] point out that it need not do so. Hence the question of global stability of equation (1) is of interest.…”

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confidence: 99%

Global stability of a biological model with time delay

Lenhart¹,

Travis²

1986

Proc. Amer. Math. Soc.

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Abstract. This paper gives necessary and sufficient conditions for global stability of certain logistic delay differential equations for all values of the delay.Biological models frequently lead to delay differential equations and to questions concerning the stability of equilibrium solutions of such models. The monographs by Cushing [6] and MacDonald [16] discuss a number of examples of such models from population dynamics, ecology, and physiology. Much of the work with such models has focused on delay equations which are reducible, through a change of variables, to systems of differential equations without delays [3,4,8,14,18,19].Another line of research is concerned with conditions under which linear retarded functional differential equations of the form L x(t) = ax(t) + £ btx(t -r,), r, > 0, /=i are asymptotically stable for all values of the delay [1,3,5,7,9,11,12,14]. We propose to combine these lines of research and examine global stability of the widely used logistic model of population dynamics, (1) x(t) -x(t) r + ax(t) 4-¿ b¡x(t -r¡)\, r, > 0, x(t) = 4>(t) for -r < t < 0, where r = max{r,}. We establish global stability (with respect to solutions with positive initial conditions) of the positive equilibrium x of equation (1). The hypothesis of our theorem guarantees that such an equilibrium exists. Koslesov [13] and Mazorov [15] point out that equation (1) (2) t\bt\.<-a i-i

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confidence: 99%

Global stability of a biological model with time delay

Lenhart¹,

Travis²

1986

Proc. Amer. Math. Soc.

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“…Since then, stage structure time delay population models of various types have been investigated by many researchers (see e.g. [7,8,9,14,15,20,21,28,32,33]).…”

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Dynamics of a two-species stage-structured model incorporating state-dependent maturation delays

Li¹,

Guo²

2017

Discrete &Amp; Continuous Dynamical Systems - B

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This paper is devoted to a cooperative model composed of two species with stage structure and state-dependent maturation delays. Firstly, positivity and boundedness of solutions are addressed to describe the population survival and the natural restriction of limited resources. It is shown that for a given pair of positive initial functions, the two mature populations are uniformly bounded away from zero and that the two mature populations are bounded above only if the the coupling strength is small enough. Moreover, if the coupling strength is large enough then the two mature populations tend to infinity as the time tends to infinity. In particular, the positivity of the two immature populations has been established under some additional conditions. Secondly, the existence and patterns of equilibria are investigated by means of degree theory and Lyapunov-Schmidt reduction. Thirdly, the local stability of the equilibria is also discussed through a formal linearization. Fourthly, the global behavior of solutions is discussed and the explicit bounds for the eventual behaviors of the two mature populations and two immature populations are obtained. Finally, global asymptotical stability is investigated by using the comparison principle of the state-dependent delay equations.

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“…Furthermore, there is a strong interaction relationship between the mature population and the immature population , which is to some extent relevant to the persistence and extinction of the related population. Consequently, it is constructive to investigate the dynamics of such ecosystem without ignorance of stage structure for population.Before 1990, the stage structure has attracted some attention from several authors [6,20,21,23,24,31,48,53,29]. The well-known biological system of single species with stage structure is established by O. Aiello and H. Freedman [1], which represents a milestone in this research field.…”

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confidence: 99%

Bifurcations and Control in a Singular Biological Economic Model

Zhang

Liu

Zhang

2012

Complexity, Analysis and Control of Singular Biological Systems

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In the natural world, there are many species whose individuals have a life history that takes them through two stages, juvenile stage and adult stage . Individuals in each stage are identical in biological characteristics, and some vital rates (rates of survival, development, and reproduction) of individuals in a population almost always depend on stage structure. Furthermore, there is a strong interaction relationship between the mature population and the immature population , which is to some extent relevant to the persistence and extinction of the related population. Consequently, it is constructive to investigate the dynamics of such ecosystem without ignorance of stage structure for population.Before 1990, the stage structure has attracted some attention from several authors [6,20,21,23,24,31,48,53,29]. The well-known biological system of single species with stage structure is established by O. Aiello and H. Freedman [1], which represents a milestone in this research field. In the following decades, much research efforts are put into the investigation of biological dynamical systems with stage structure for population. Some related research results can be found in [1,2,3,10,14,26,8,39,34,35,36,37,38,40,44,41,42,43,45,47,51,52,54,25,16,9] and the references therein.Recently, the dynamics of a class of stage-structured prey-predator models with discrete time delay have been studied by several authors [4,55,18,58,5] and the references therein.Arino et al. [4] suggested that the time delay to adulthood should be state dependent and careful formulation of such state dependent time delays can lead to models that produce periodic solutions. Xu et al. [55] studied the persistence and stability of a delayed prey-predator model with stage structure for predator. Gourley et al. [18] formulated a general and robust prey predator model with stage-structure with constant maturation time delay and performed a systematic mathematical and computational Q. Zhang et al.: Complex

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Properties of Solutions of a Class of Equations With Lag Which Describe the Dynamics of Change in the Population of a Species With the Age Structure Taken Into Account

Cited by 14 publications

References 5 publications

Global stability of a biological model with time delay

Global stability of a biological model with time delay

Dynamics of a two-species stage-structured model incorporating state-dependent maturation delays

Bifurcations and Control in a Singular Biological Economic Model

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