Global qualitative analysis of a discrete host‐parasitoid model with refuge and strong Allee effects

In this paper, we discuss the qualitative behavior of a discrete host‐parasitoid model with the host subject to refuge and strong Allee effects. More precisely, we study the local and global asymptotic stability, stable manifolds and unstable manifolds of boundary equilibrium points, existence and unique positive equilibrium point, local and global behavior of the positive equilibrium point, and the uniform persistence for the model with the host subject to the refuge or both refuge and strong Allee effects. It is also proved that the model undergoes a transcritical bifurcation in a small neighborhood of the boundary equilibrium point. Some numerical simulations are given to support our theoretical results. We can obtain that the addition of the refuge may make the parasitoids go extinct while the hosts survive or may stabilize the host‐parasitoid interaction; the addition of both refuge and strong Allee effects has either a negative or positive impact on the coexistence of both populations.     

OPTIMAL INSTAR PARASITIZATION IN A STAGE STRUCTURED HOST-PARASITOID MODEL

A stage structured host-parasitoid model is derived and the equilibria studied. It is shown under what conditions the parasitoid controls an exponentially growing host in the sense that a coexistence equilibrium exists. Furthermore, for host populations whose inherent growth rate is not too large it is proved that in order to minimize the adult host equilibrium level it is necessary that the parasitoids attack only one of the larval stages. It is also proved in this case that the minimum adult host equilibrium level is attained when the parasitoids attack that larval stage which also maximizes the expected number of emerging adult parasitoid per larva at equilibrium. Numerical simulations tentatively indicate that the first conclusion remains in general valid for the model. However, numerical studies also show that it is not true in general that the optimal strategy will maximize the number of emerging adult parasitoid per larva at equilibrium.     

Discrete-time host–parasitoid models with Allee effects: Density dependence versus parasitism

We present two general discrete-time host–parasitoid models with Allee effects on the host. In the first model, it is assumed that parasitism occurs prior to density dependence, while in the second model we assume that density dependence operates first followed by parasitism. It is shown that both models have similar asymptotic behaviour. The parasitoid population will definitely go extinct if the maximal growth rate of the host population is less than or equal to one, independent of whether density dependence or parasitism occurs first. The fate of the population is initial condition dependent if this maximal growth rate exceeds one. In particular, there exists a host population threshold, the Allee threshold, below which the host population goes extinct and so does the parasitoid. This threshold is the same for both models. Numerical examples with different functions are simulated to illustrate our analytical results.     

COMBINING METHODS OF PEST CONTROL: COMPLEMENTARITY OF METHODS AND A GUIDING PRINCIPLE

Four sets of models are examined which represent various pairwise combinations of several methods of pest control. These methods involve the release of sterile male pests, the inundative release of parasitoids, insecticide application, pheromone trapping and food-baited trapping with either insecticides or sterilants. It was observed that two pest control methods will combine synergistically, and thus be complimentary, if their optimal action is at different pest densities and varies differently with pest density. The synergism thus generated by differing dependence on density, can however, be obscured if the two control methods interfere with each other in some other way, as occurs for example with the use of both insecticides and inundative release of parasitoids.     

POPULATION AND HARVESTING THEORY FOR NONLINEAR SEX AND AGE-STRUCTURED RESOURCES

A sex-age-structured population model with density dependence in the conversion of reproductive potentials into zygotes and in first year survivorship is described. The model has two equilibria; the smallest is mathematically unstable, and the origin and the larger equilibrium are locally stable. The population can thus go extinct for certain initial states, or if the two equilibria coincide. The ratio between the two equilibria can be regarded as a measure of the risk of extinction, since it is related to the chance that detrimental environmental conditions will cause the population to enter the region of attraction of the origin. In simple monoecious models, recovery to former levels is only possible provided that the population is not driven to extinction before harvesting effort is reduced. Ratios between the two unexploited equilibria, and between the stable unexploited equilibrium and the recruitment level at which the two equilibria coincide are given solely in terms of the degree of density dependence in the model. I show that the harvesting strategy which maximizes the equilibrium yield has a four age form, involving harvesting of at most two male and two female age classes. Out of ten commercial Pacific groundfish species, knife-edge selectivity sustainable yields of eight are at least 90% of ultimate sustainable yield (USY). With no effort restrictions, the range of lengths at first capture which achieve more than 60% of USY is narrow. When one of the sexes is not harvested, sustainable yield is between 20% and 80% of USY, but lowest when females are not harvested.     

JOINTLY‐DETERMINED ECOLOGICAL THRESHOLDS AND ECONOMIC TRADE‐OFFS IN WILDLIFE DISEASE MANAGEMENT

ABSTRACT. We investigate wildlife disease management, in a bioeconomic framework, when the wildlife host is valuable and disease transmission is density‐dependent. Disease prevalence is reduced in density‐dependent models whenever the population is harvested below a host‐density threshold a threshold population density below which disease prevalence declines and above which a disease becomes epidemic. In conventional models, the threshold is an exogenous function of disease parameters. We consider this case and find a steady state with positive disease prevalence to be optimal. Next, we consider a case in which disease dynamics are affected by both population controls and changes in human‐environmental interactions. The host‐density threshold is endogenous in this case. That is, the manager does not simply manage the population relative to the threshold, but rather manages both the population and the threshold. The optimal threshold depends on the economic and ecological trade‐offs arising from the jointly‐determined system. Accounting for this endogene‐ity can lead to reduced disease prevalence rates and higher population levels. Additionally, we show that ecological parameters that may be unimportant in conventional models that do not account for the endogeneity of the host‐density threshold are potentially important when host density threshold is recognized as endogenous.     

Dynamics of an age‐structured host‐vector model for malaria transmission

In this paper, we propose a host‐vector model for malaria transmission by incorporating infection age in the infected host population and nonlinear incidence for transmission from infectious vectors to susceptible hosts. One novelty of the model is that the recovered hosts only have temporary immunity and another is that successfully recovered infected hosts may become susceptible immediately. Firstly, the existence and local stability of equilibria is studied. Secondly, rigorous mathematical analyses on technical materials and necessary arguments, including asymptotic smoothness and uniform persistence of the system, are given. Thirdly, by applying the fluctuation lemma and the approach of Lyapunov functionals, the threshold dynamics of the model for a special case were established. Roughly speaking, the disease‐free equilibrium is globally asymptotically stable when the basic reproduction number is less than one and otherwise the endemic equilibrium is globally asymptotically stable when no reinfection occurs. It is shown that the infection age and nonlinear incidence not only impact on the basic reproduction number but also could affect the values of the endemic steady state. Numerical simulations were performed to support the theoretical results.     

ALLEE EFFECTS: POPULATION GROWTH,CRITICAL DENSITY,AND THE CHANCE OF EXTINCTION

This paper develops mathematical models to describe the growth, critical density, and extinction probability in sparse populations experiencing Allee effects. An Allee effect (or depensation) is a situation at low population densities where the per-individual growth rate is an increasing function of population density. A potentially important mechanism causing Allee effects is a shortage of mating encounters in sparse populations. Stochastic models are proposed for predicting the probability of encounter or the frequency of encounter as a function of population density. A negative exponential function is derived as such an encounter function under very general biological assumptions, including random, regular, or aggregated spatial patterns. A rectangular hyperbola function, heretofore used in ecology as the functional response of predator feeding rate to prey density, arises from the negative exponential function when encounter probabilities are assumed heterogeneous among individuals. These encounter functions produce Allee effects when incorporated into population growth models as birth rates. Three types of population models with encounter-limited birth rates are compared: (1) deterministic differential equations, (2) stochastic discrete birth-death processes, and (3) stochastic continuous diffusion processes. The phenomenon of a critical density, a major consequence of Allee effects, manifests itself differently in the different types of models. The critical density is a lower unstable equilibrium in the deterministic differential equation models. For the stochastic discrete birth-death processes considered here, the critical density is an inflection point in the probability of extinction plotted as a function of initial population density. In the continuous diffusion processes, the critical density becomes a local minimum (antimode) in the stationary probability distribution for population density. For both types of stochastic models, a critical density appears as an inflection point in the probability of attaining a small population density (extinction) before attaining a large one. Multiplicative (“environmental”) stochastic noise amplifies Allee effects. Harvesting also amplifies those effects. Though Allee effects are difficult to detect or measure in natural populations, their presence would seriously impact exploitation, management, and preservation of biological resources.     

A MODEL OF THE ENZOOTIOLOGY OF LYME DISEASE IN THE ATLANTIC NORTHEAST OF THE UNITED STATES

A mathematical model is presented for the dynamics of the rate of infection of the Lyme disease vector tick Ixodes dammini (Acari: Ixodidae) by the spirochete Borrelia burgdorferi, in the Atlantic Northeast of the United States. According to this model, moderate reductions in the abundance of white-tailed deer Odocoileus virginianus may either decrease or increase the spirochete infection rate in ticks, provided the deer are not reservoir hosts for Lyme disease. Expressions for the basic reproductive rate of the disease are computed analytically for special cases, and it is shown that as the basic reproductive rate increases, a proportional reduction in the tick population produces a smaller proportional reduction in the infection rate, so that vector control is less effective far above the threshold. The model also shows that control of the mouse reservoir hosts Peromyscus leucopus could reduce the infection rate if the survivorship of juvenile stages of ticks were reduced as a consequence. If the survivorship of juvenile stages does not decline as the rodent population is reduced, then rodent reduction can increase the spirochete infection rate in the ticks.     

KAPPA FUNCTION AS A UNIFYING FRAMEWORK FOR DISCRETE POPULATION MODELING

This paper develops a unified way to describe the various generalized discrete‐time nonlinear dynamical models with density dependence, Allee effects, and parasitoids. We show how the kappa function can be used to describe the probabilities involved in intra‐ or interspecific encounters, namely, (i) the probability of surviving to the next generation in the absence of parasitoids or Allee effects, (ii) the encounter probability associated with Allee effects, and (iii) the probability of escaping parasitism in the presence of parasitoids. Having introduced a phenomenological framework of modeling via the kappa function, we then provide a realistic mechanism through stochastic encounters, responsible for generating the kappa function to any of the three involved probabilities. The unified modeling through the kappa function yields insights into how abundances influence species interactions. It is now straightforward to use this unified modeling to analyze and investigate its consequences in species dynamics.     

USING RESERVES TO PROTECT FISH AND WILDLIFE SIMPLIFIED MODELING APPROACHES

ABSTRACT. This paper investigates theoretically to what extent a nature reserve may protect a uniformly distributed population of fish or wildlife against negative effects of harvesting. Two objectives of this protection are considered: avoidance of population extinction and maintenance of population, at or above a given precautionary population level. The pre‐reserve population is assumed to follow the logistic growth law and two models for post‐reserve population dynamics are formulated and discussed. For Model A by assumption the logistic growth law with a common carrying capacity is valid also for the post‐reserve population growth. In Model B, it is assumed that each sub‐population has its own carrying capacity proportionate to its distribution area. For both models, migration from the high‐density area to the low‐density area is proportional to the density difference. For both models there are two possible outcomes, either a unique globally stable equilibrium, or extinction. The latter may occur when the exploitation effort is above a threshold that is derived explicitly for both models. However, when the migration rate is less than the growth rate both models imply that the reserve can be chosen so that extinction cannot occur. For the opposite case, when migration is large compared to natural growth, a reserve as the only management tool cannot assure survival of the population, but the specific way it increases critical effort is discussed.     

Maintenance of diversity in a hierarchical host–parasite model with balancing selection and reinfection

Inspired by DNA data of the human cytomegalovirus we propose a model of a two-type parasite population distributed over its hosts. The parasite is capable to persist in its host till the host dies, and to reinfect other hosts. To maintain type diversity within a host, balancing selection is assumed.For a suitable parameter regime we show that in the limit of large host and parasite populations the host state frequencies follow a dynamical system with a globally stable equilibrium, guaranteeing that both types are maintained in the parasite population for a long time on the host time scale.     

The joint mortality of couples in continuous time

This paper introduces a probabilistic framework for the joint survivorship of couples in the context of dynamic stochastic mortality models. The death of one member of a couple can have either deterministic or stochastic effects on the other; our new framework gives an intuitive and flexible pairwise cohort-based probabilistic mechanism that can account for both. It is sufficiently flexible to allow modelling of effects that are short-term (called the broken-heart effect) and/or long-term (named life circumstances bereavement). In addition, it can account for the state of health of both the surviving and the dying spouse and can allow for dynamic and asymmetric reactions of varying complexity. Finally, it can accommodate the pairwise dependence of mortality intensities before the first death. Analytical expressions for bivariate survivorship in representative models are given, and their sensitivity analysis is performed for benchmark cases of old and young couples. Simulation and estimation procedures are provided that are straightforward to implement and lead to consistent parameter estimation on synthetic dataset of 10000 pairs of death times for couples.     

Effects of environmental fluctuation and time delay on ratio dependent hyperparasitism model

A model of host–parasitoid–hyperparasitoid is considered with ratio dependence between parasitoid and hyperparasitoid. First, the conditions for local stability and increasing host fitness due to the effect of hyperparasitism are deduced. Next, we study the effects of stochastic environmental fluctuations and discrete time delay on the system behavior and calculate the corresponding populations variances. Numerical simulations illustrate that populations densities oscillate randomly around equilibrium points. Also, in contrast to previous literature, the simulations carried out here indicate that populations variances oscillate with the increase of time delay.     

Neimark-Sacker bifurcation and chaos control in Hassell-Varley model

We investigate the complex behaviour of a modified Nicholson–Bailey model. The modification is proposed by Hassel and Varley taking into account that interaction between parasitoids is taken in such a way that the searching area per parasitoid is inversely proportional to the m-th power of the population density of parasitoids. Under certain parametric conditions the unique positive equilibrium point of system is locally asymptotically stable. Moreover, it is proved that system undergoes Neimark-Sacker bifurcation for small range of parameters by using standard mathematical techniques of bifurcation theory. In order to control Neimark-Sacker bifurcation, we apply simple feedback control strategy and pole-placement technique which is a modification of OGY method. Moreover, the hybrid control methodology is also implemented for chaos controlling. Numerical simulations are provided to illustrate theoretical discussion.     

Stability Switches in a Host–Pathogen Model as the Length of a Time Delay Increases

The destabilising effects of a time delay in mathematical models are well known. However, delays are not necessarily destabilising. In this paper, we explore an example of a biological system where a time delay can be both stabilising and destabilising. This example is a host–pathogen model, incorporating density-dependent prophylaxis (DDP). DDP describes when individual hosts invest more in immunity when population densities are high, due to the increased risk of infection in crowded conditions. In this system, as the delay length increases, there are a finite number of switches between stable and unstable behaviour. These stability switches are demonstrated and characterised using a combination of numerical methods and analysis.     

Modeling the effects of variable external influences and demographic processes on innovation diffusion

In this paper, a nonlinear mathematical model for innovation diffusion is proposed and analyzed by considering the effects of variable external influences (cumulative marketing efforts) and human population (variable marketing potential) in a society. The change in the population density is caused by various demographic processes such as immigration, emigration, intrinsic growth rate, death rate, etc.Thus, the problem of innovation diffusion is governed by three dynamic variables, namely, non adopters’ density, adopters’ density and the cumulative density of external influences. The model is analyzed by using the stability theory of differential equations and computer simulation.The model analysis shows that the main effect of the increase in cumulative density of external influences is to make the adopter population density reach its equilibrium at a much faster rate. It further shows that the density of adopters’ population increases as the parameters related to increase in non adopters’ population density increase. The effects of various parameters in the model on the nature of existing single equilibrium have also been discussed by using numerical simulation. It is shown that parameters related to the growth of non adopters’ population density have stabilizing effects on the system.     

A CLASS OF MODELS DESCRIBING AGE STRUCTURE DYNAMICS IN A NATURAL FOREST

ABSTRACT. The age dynamics of a natural forest is modeled by the von‐Foerster partial differential equation for the age density, while the seedling density is obtained as a solution of an integro‐differential equation. This seedling density equation contains a small parameter, the ratio of seedling re‐establishment time and the life span of an average tree in the forest. Several models are introduced that take into account various mortality curves and growth functions of trees, the dependence of seedlings carrying capacity on forest size, and different types of seedlings re‐establishment. Asymptotic, analytic and numerical methods are used to solve typical example problems.     

Harvesting discrete nonlinear age and stage structured populations

A natural extension of age structured Leslie matrix models is to replace age classes with stage classes and to assume that, in each time period, the transition from one stage class to the next is incomplete; that is, diagonal terms appear in the transition matrix. This approach is particularly useful in resource systems where size is more easily measured than age. In this linear setting, the properties of the models are known; and these models have been applied to the analysis of population problems. A more applicable setting is to assume that the reproduction, survival, and transition parameters in the model are density dependent. The behavior of such models is determined by the form of this density dependence. Here, we focus on models in which the parameters depend on the value of an aggregated variable, defined to be the weighted sum of the number of individuals in each stage class. In forestry models, for example, this aggregated variable may represent a basal area index; in fisheries models, it may represent a spawning stock biomass. Current age structured nonlinear stock-recruitment fisheries models are a special case of the models considered here. Certain results that apply to age structured models can be extended to this broader class of models. In particular, the questions addressed relate to the minimum number of age classes that need to be harvested to obtain maximum sustainable yield policies and to managing resources under nonequilibrium and stochastic conditions. Application of the model to problems in fisheries, forestry, pest, and wildlife management is also discussed.The author would like to thank R. G. Haight for comments and discussions relating to the material presented here. This work was supported by NSF Grant DMS-85-11717.