Stability of an age specific population with density dependent fertility

A nonlinear version of the Lotka-Sharpe model of population growth is considered in which the age specific fertility is a function of the population size. The stability of an equilibrium population distribution is investigated with respect to both global and local perturbations. Sufficient conditions for such stability are presented, as are estimates for the rate of return of the population to the equilibrium configuration. Particular attention is paid to those situations in which the age dependent stability criteria coincide with those of age independent models.     

Existence and stability of equilibria in age-structured population dynamics

The existence of positive equilibrium solutions of the McKendrick equations for the dynamics of an age-structured population is studied as a bifurcation phenomenon using the inherent net reproductive rate n as a bifurcation parameter. The local existence and uniqueness of a branch of positive equilibria which bifurcates from the trivial (identically zero) solution at the critical value n=1 are proved by implicit function techniques under very mild smoothness conditions on the death and fertility rates as functional of age and population density. This first requires the development of a suitable linear theory. The lowest order terms in the Liapunov-Schmidt expansions are also calculated. This local analysis supplements earlier global bifurcation results of the author. The stability of both the trivial and the positive branch equilibria is studied by means of the principle of linearized stability. It is shown that in general the trivial solution losses stability as n increases through one while the stability of the branch solution is stable if and only if the bifurcation is supercritical. Thus the McKendrick equations exhibit, in the latter case, a standard exchange of stability with regard to equilibrium states as they depend on the inherent net reproductive rate. The derived lower order terms in the Liapunov-Schmidt expansions yield formulas which explicitly relate the direction of bifurcation to properties of the age-specific death and fertility rates as functionals of population density. Analytical and numerical results for some examples are given which illustrate these results.     

Local stability of a population with density-dependent fertility

The local stability of an equilibrium population configuration is investigated. The population is governed by a density-dependent, age-structured, nonlinear renewal equation. Local stability is shown to depend essentially on the rate of change of the net reproduction rate as a function of the population size. Examples illustrating the results and the characteristics promoting stability are discussed.     

The role of seasonality in the dynamics of deer tick populations

In this paper, we formulate a nonlinear system of difference equations that models the three-stage life cycle of the deer tick over four seasons. We study the effect of seasonality on the stability and oscillatory behavior of the tick population by comparing analytically the seasonal model with a non-seasonal one. The analysis of the models reveals the existence of two equilibrium points. We discuss the necessary and sufficient conditions for local asymptotic stability of the equilibria and analyze the boundedness and oscillatory behavior of the solutions. A main result of the mathematical analysis is that seasonality in the life cycle of the deer tick can have a positive effect, in the sense that it increases the stability of the system. It is also shown that for some combination of parameters within the stability region, perturbations will result in a return to the equilibrium through transient oscillations. The models are used to explore the biological consequences of parameter variations reflecting expected environmental changes.     

Models for the spread of universally fatal diseases

In the formulation of models of S-I-R type for the spread of communicable diseases it is necessary to distinguish between diseases with recovery with full immunity and diseases with permanent removal by death. We consider models which include nonlinear population dynamics with permanent removal. The principal result is that the stability of endemic equilibrium may depend on the population dynamics and on the distribution of infective periods; sustained oscillations are possible in some cases.     

Structured population dynamics: continuous size and discontinuous stage structures

A nonlinear stochastic model for the dynamics of a population with either a continuous size structure or a discontinuous stage structure is formulated in the Eulerian formalism. It takes into account dispersion effects due to stochastic variability of the development process of the individuals. The discrete equations of the numerical approximation are derived, and an analysis of the existence and stability of the equilibrium states is performed. An application to a copepod population is illustrated; numerical results of Eulerian and Lagrangian models are compared.      

A stage-structured predator–prey model with distributed maturation delay and harvesting

ABSTRACT

A stage-structured predator–prey system with distributed maturation delay and harvesting is investigated. General birth and death functions are used. The local stability of each feasible equilibria is discussed. By using the persistence theory, it is proven that the system is permanent if the coexistence equilibrium exists. By using Lyapunov functional and LaSalle invariant principle, it is shown that the trivial equilibrium is globally stable when the other equilibria are not feasible, and that the boundary equilibrium is globally stable if the coexistence equilibrium does not exist. Finally, sufficient conditions are derived for the global stability of the coexistence equilibrium.     

具有种群Logistic增长的SIR模型的稳定性和Hopf分支

基于种群Logistic增长假设,本文构建了一个三维非线性发生率的SIR模型.首先探讨了该模型平衡点的稳定性,然后应用中心流形投影法得到了系统在非平凡平衡点附近产生超临界分支,并给出了数值模拟.     

Regions of Stable Equilibria for Models of Differential Selection in the Two Sexes under Random Mating

The equilibrium structure of models of differential selection in the sexes is investigated. It is shown that opposing additive selection leads to stable polymorphic equilibria for only a restricted set of selection intensities, and that for weak selection the selection intensities must be of approximately the same magnitude in the sexes. General models of opposing directional selection, with arbitrary dominance, are investigated by considering simultaneously the stability properties of the trivial equilibria and the curve along which multiple roots appear. Numerical calculations lead us to infer that the average degree of dominance determines the equilibrium characteristics of models of opposing selection. It appears that if the favored alleles are, on the average, recessive, there may be multiple polymorphic equilibria, whereas only a single polymorphic equilibrium can occur when the favored alleles are, on the average, dominant. The principle that the average degree of dominance controls equilibrium behavior is then extended to models allowing directional selection in one sex with overdominance in the other sex, by showing that polymorphism is maintained if and only if the average fitness in heterozygotes exceeds one.     

Lotka-Volterra equations: Decomposition,stability, and structure

Summary The major objective of this paper is to propose a new decomposition-aggregation framework for stability analysis of Lotka-Volterra equations employing the concept of vector Liapunov functions. Both the disjoint and the overlapping decompositions are introduced to increase flexibility in constructing Liapunov functions for the overall system. Our second objective is to consider the Lotka-Volterra equations under structural perturbations, and derive conditions under which a positive equilibrium is connectively stable. Both objectives of this paper are directed towards a better understanding of the intricate interplay between stability and complexity in the context of robustness of model ecosystems represented by Lotka-Volterra equations. Only stability of equilibria in models with constant parameters is considered here. Nonequilibrium analysis of models with nonlinear time-varying parameters is the subject of a companion paper.Research supported by U.S. Department of Energy under the Contract EC-77-S-03-1493.On leave from Kobe University, Kobe, Japan.     

Fast and Slow dynamics of Malaria model with relapse

Two mathematical models of malaria with relapse are studied. When the vector population size is constant, complete analyses of the dynamics are conducted. The geometric singular perturbation theory is used to analyze the full dynamics. On the critical manifold, from next generation matrix method, we obtain the basic reproduction number. The global stability of disease-free equilibrium and the uniformly persistence of malaria have also been analyzed. While the vector population size is variable, the basic reproduction number and the stability of disease-free as well as the malaria-infected equilibrium have been obtained in a similar way. Some numerical simulations are also given.     

Will small population sizes warn us of impending extinctions?

Several models are used to show that population sizes are often relatively insensitive to deteriorating environmental conditions over most of the range of environments that allow population persistence. As conditions continue to worsen in these cases, equilibrium population sizes ultimately decline rapidly toward extinction from sizes similar to or larger than those before environmental decline began. Consumer-resource models predict that equilibrium or average population size can increase with deteriorating environmental conditions over a large part of the range of the environmental parameter that allows persistence. The initial insensitivity or increase in the population of the focal species occurs because changes in the populations of other components of the food web compensate for the decline in one or more fitness components of the focal population. However, the compensatory processes are generally nonlinear and often approach their limits abruptly rather than gradually. When there is steady directional change in the environment, populations lag behind the equilibrium population size specified by current environmental conditions. The environmental variable can then decline below the level required for population persistence while the population size is still close to or greater than its original value. Efficient consumers and self-reproducing resources are especially likely to produce this outcome. More complex models with adaptive behavior, additional consumers, or additional resources often exhibit similar trajectories of population size under environmental deterioration.     

Disease transmission models with density-dependent demographics

The models considered for the spread of an infectious disease in a population are of SIRS or SIS type with a standard incidence expression. The varying population size is described by a modification of the logistic differential equation which includes a term for disease-related deaths. The models have density-dependent restricted growth due to a decreasing birth rate and an increasing death rate as the population size increases towards its carrying capacity. Thresholds, equilibria and stability are determined for the systems of ordinary differential equations for each model. The persistence of the infectious disease and disease-related deaths can lead to a new equilibrium population size below the carrying capacity and can even cause the population to become extinct.Research supported in part by Centers for Disease Control contract 200-87-0515     

Equilibrium and Nonequilibrium Models in Ecological Anthropology: An Evaluation of "Stability" in Maring Ecosystems in New Guinea

Three models pertaining to the stability of Maring ecosystems have been proposed. The first is the local stability model, in which a population seeks its own equilibrium state; the second is the regional stability model, in which each population is ultimately unstable, but populations persist somewhere in space; and the third is the disequilibrium model, in which neither stability nor population regulation is attained. In the disequilibrium model, exogenous factors prevent a population, which is moving toward some equilibrium state, from reaching it. The large number of quantitative anthropological and ecological studies in Highlands New Guinea has not shown clearly which of these three models best describes reality. Simulation of shifting agriculture in New Guinea shows that the Highlands systems are equilibrium-seeking, but have such limited recovery rates from disturbance that even small perturbations are sufficient to keep them from reaching equilibrium. When the influences of technological innovation, environmental change, and social-cultural evolution are taken into account, the disequilibrium model is the model of choice. These systems remain away from their stable equilibrium points most of the time, if those exist at all. Thus, New Guinea agroecosystems can be stable or unstable depending upon how stability is defined.     

The impact of mortality on predator population size and stability in systems with stage-structured prey

The relationships between a predator population's mortality rate and its population size and stability are investigated for several simple predator-prey models with stage-structured prey populations. Several alternative models are considered; these differ in their assumptions about the nature of density dependence in the prey's population growth; the nature of stage-transitions; and the stage-selectivity of the predator. Instability occurs at high, rather than low predator mortality rates in most models with highly stage-selective predation; this is the opposite of the effect of mortality on stability in models with homogeneous prey populations. Stage-selective predation also increases the range of parameters that lead to a stable equilibrium. The results suggest that it may be common for a stable predator population to increase in abundance as its own mortality rate increases in stable systems, provided that the predator has a saturating functional response. Sufficiently strong density dependence in the prey generally reverses this outcome, and results in a decrease in predator population size with increasing predator mortality rate. Stability is decreased when the juvenile stage has a fixed duration, but population increases with increasing mortality are still observed in large areas of stable parameter space. This raises two coupled questions which are as yet unanswered; (1) do such increases in population size with higher mortality actually occur in nature; and (2) if not, what prevents them from occurring? Stage-structured prey and stage-related predation can also reverse the 'paradox of enrichment', leading to stability rather than instability when prey growth is increased.     

具有非线性种内制约关系和分布时滞竞争模型的稳定性

构造并研究了一类具有分布时滞和非线性种内制约关系的竞争模型.得到了这类模型的边界平衡点和共存平衡点全局渐近稳定的条件,以及分布时滞、非线性种内制约关系对模型的影响.     

Pest management through continuous and impulsive control strategies

In this paper, we propose two mathematical models concerning continuous and, respectively, impulsive pest control strategies. In the case in which a continuous control is used, it is shown that the model admits a globally asymptotically stable positive equilibrium under appropriate conditions which involve parameter estimations. As a result, the global asymptotic stability of the unique positive equilibrium is used to establish a procedure to maintain the pests at an acceptably low level in the long term. In the case in which an impulsive control is used, it is observed that there exists a globally asymptotically stable susceptible pest-eradication periodic solution on condition that the amount of infective pests released periodically is larger than some critical value. When the amount of infective pests released is less than this critical value, the system is shown to be permanent, which implies that the trivial susceptible pest-eradication solution loses its stability. Further, the existence of a nontrivial periodic solution is also studied by means of numerical simulation. Finally, the efficiency of continuous and impulsive control policies is compared.     

Global analysis on delay epidemiological dynamic models with nonlinear incidence

In this paper, we derive and study the classical SIR, SIS, SEIR and SEI models of epidemiological dynamics with time delays and a general incidence rate. By constructing Lyapunov functionals, the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium is shown. This analysis extends and develops further our previous results and can be applied to the other biological dynamics, including such as single species population delay models and chemostat models with delay response.     

Some threshold and stability results for epidemic models with a density-dependent death rate.

Most classical models for infectious diseases assume that the birth and death rates of individuals and the meeting rates between susceptible and infected individuals do not depend on the total number of individuals in the population. While these assumptions are valid in some situations they are less valid in others. For example, for diseases in animal an insects populations competition for scarce resources might well mean that the death rate depends on the number of individuals. The present paper examines two epidemic models where the death rate is density dependent. For each model the possible equilibrium levels of disease incidence are determined and the stability of these equilibrium levels to small perturbations is discussed. The biological interpretation of these results is presented together with the results of some numerical simulations.