Threshold of disease transmission in a patch environment

An epidemic model is proposed to describe the dynamics of disease spread between two patches due to population dispersal. It is proved that reproduction number is a threshold of the uniform persistence and disappearance of the disease. It is found that the dispersal rates of susceptible individuals do not influence the persistence and extinction of the disease. Furthermore, if the disease becomes extinct in each patch when the patches are isolated, the disease remains extinct when the population dispersal occurs; if the disease spreads in each patch when the patches are isolated, the disease remains persistent in two patches when the population dispersal occurs; if the disease disappears in one patch and spreads in the other patch when they are isolated, the disease can spread in all the patches or disappear in all the patches if dispersal rates of infectious individuals are suitably chosen. It is shown that an endemic equilibrium is locally stable if susceptible dispersal occurs and infectious dispersal turns off. If susceptible individuals and infectious individuals have the same dispersal rate in each patch, it is shown that the fractions of infectious individuals converge to a unique endemic equilibrium.     

Evolution of dispersal in a spatially periodic integrodifference model

An integrodifference model describing the reproduction and dispersal of a population is introduced to investigate the evolution of dispersal in a spatially periodic habitat. The dispersal is determined by a kernel function, and the dispersal strategy is defined as the probability of population individuals’ moving to a different habitat. Both conditional and unconditional dispersal strategies are investigated, the distinction being whether dispersal depends on local environmental conditions. For competing unconditional dispersers, we prove that the population with the smaller dispersal probability always prevails. Alternatively, for conditional dispersers, it is shown that the strategy known as ideal free dispersal is both sufficient and necessary for evolutionary stability. These results extend those in the literature for discrete diffusion models in finite patchy landscapes and from reaction–diffusion models.     

Harvesting control in an integrodifference population model with concave growth term

We consider the harvest of a certain proportion of a population that is modeled by an integrodifference equation, which is discrete in time and continuous in the space variable. The dispersal of the population is modeled by an integral of the growth function evaluated at the current population density against a kernel function. A concave growth function is used. In our model, growth occurs first, then dispersal and lastly harvesting control before the next generation. With the goal of maximizing the discounted profit stream, the optimal control is characterized by an optimality system. Illustrative examples are computed numerically.     

Permanence of dispersal population model with time delays

Permanence of a dispersal single-species population model is considered where environment is partitioned into several patches and the species requires some time to disperse between the patches. The model is described by delay differential equations. The existence of food-rich patches and small dispersions among the patches are proved to be sufficient to ensure partial permanence of the model. It is also shown that partial permanence ensures permanence if each food-poor patch is connected to at least one food-rich patch and if each pair in food-rich patches is connected. Furthermore, it is proved that partial persistence is ensured even under large dispersion among food-rich patches if the dispersion time is relatively small.     

A METAPOPULATION MODEL WITH DISCRETE SIZE STRUCTURE

ABSTRACT. We consider a discrete size‐structured meta‐population model with the proportions of patches occupied by n individuals as dependent variables. Adults are territorial and stay on a certain patch. The juveniles may emigrate to enter a dispersers' pool from which they can settle on another patch and become adults. Absence of colonization and absence of emigration lead to extinction of the metapopula‐tion. We define the basic reproduction number R₀ of the metapopulation as a measure for its strength of persistence. The metapopulation is uniformly weakly persistent if R₀> 1. We identify subcritical bifurcation of persistence equilibria from the extinction equilibrium as a source of multiple persistence equilibria: it occurs, e.g., when the immigration rate, into occupied patches, exceeds the colonization rate (of empty patches). We determine that the persistence‐optimal dispersal strategy which maximizes the basic reproduction number is of bang‐bang type: If the number of adults on a patch is below carrying capacity all the juveniles should stay, if it is above the carrying capacity all the juveniles should leave.     

A bifurcation analysis of stage-structured density dependent integrodifference equations

There is evidence for density dependent dispersal in many stage-structured species, including flour beetles of the genus Tribolium. We develop a bifurcation theory approach to the existence and stability of (non-extinction) equilibria for a general class of structured integrodifference equation models on finite spatial domains with density dependent kernels, allowing for non-dispersing stages as well as partial dispersal. We show that a continuum of such equilibria bifurcates from the extinction equilibrium when it loses stability as the net reproductive number n increases through 1. Furthermore, the stability of the non-extinction equilibria is determined by the direction of the bifurcation. We provide an example to illustrate the theory.     

The diffusive Lotka–Volterra competition model in fragmented patches I: Coexistence

It is an ecological imperative that we understand how changes in landscape heterogeneity affect population dynamics and coexistence among species residing in increasingly fragmented landscapes. Decades of research have shown the dispersal process to have major implications for individual fitness, species’ distributions, interactions with other species, population dynamics, and stability. Although theoretical models have played a crucial role in predicting population level effects of dispersal, these models have largely ignored the conditional dependency of dispersal (e.g., responses to patch boundaries, matrix hostility, competitors, and predators). This work is the first in a series where we explore dynamics of the diffusive Lotka–Volterra (L–V) competition model in such a fragmented landscape. This model has been extensively studied in isolated patches, and to a lesser extent, in patches surrounded by an immediately hostile matrix. However, little attention has been focused on studying the model in a more realistic setting considering organismal behavior at the patch/matrix interface. Here, we provide a mechanistic connection between the model and its biological underpinnings and study its dynamics via exploration of nonexistence, existence, and uniqueness of the model’s steady states. We employ several tools from nonlinear analysis, including sub-supersolutions, certain eigenvalue problems, and a numerical shooting method. In the case of weak, neutral, and strong competition, our results mostly match those of the isolated patch or immediately hostile matrix cases. However, in the case where competition is weak towards one species and strong towards the other, we find existence of a maximum patch size, and thus an intermediate range of patch sizes where coexistence is possible, in a patch surrounded by an intermediate hostile matrix when the weaker competitor has a dispersal advantage. These results support what ecologists have long theorized, i.e., a key mechanism promoting coexistence among competing species is a tradeoff between dispersal and competitive ability.     

The Effect of Dispersal on Population Growth with Stage-structure

The effect of dispersal on the permanence of population in a polluted patch is studied in this paper. The authors constructed a single-species dispersal model with stage-structure in two patches. The analysis focuses on the case that the toxicant input in the polluted patch has a limit value. The authors derived the conditions under which the population will be either permanent, or extinct.     

Spatially explicit ecological models: a spatial convolution approach

Spatial structure tends to have a stabilizing influence on predator–prey interactions in which the local model predicts extinction of the system. This result is well supported by laboratory observations of simple systems. Here, we use a spatially explicit version of the Nicholson–Bailey model having Moran–Ricker host reproduction to repeat and extend some of these results. Our model is a discrete spatial convolution model analogous to the integrodifference equations (IDEs) used by other authors. We show a spatial rescue effect which prevents extinction of the system by reducing the size (standard deviation) of the dispersal pdf. We also show that very favorable habitat (K=∞) and marginal habitat (K=1.0), when mixed randomly together in an explicit map, are highly stabilizing whereas either kind of habitat alone will cause extinction. The marginal habitat in this situation has host densities below parasite replacement level and thus constitutes a host refuge (although not a complete one) from the parasite. When a host–parasitoid model having spiral wave dynamics in two-dimensional space was extended to one- and three-dimensional space, we observed analogous dynamics, i.e., traveling waves of evasion and pursuit in one dimension and ‘spiral-like’ structures in a three-dimensional spatial volume. We illustrate an approach to analysis of spatial convolution models via the frequency response of the system transfer function. In spatial convolution format, local interaction and dispersal are conveniently isolated from one another, and this allows us to vary these components independently and thus to study their effects on the dynamics of the total system. We show two examples of nonrandom dispersal pdf’s – a bimodal form representing two dispersal types in the population and a ‘ripple’ pdf representing a repulsive process.     

PERSISTENCE IN AN AGE-STRUCTURED POPULATION FOR A PATCH-TYPE ENVIRONMENT

A discrete-time model for an age-structured population in a patch-type environment is presented and analyzed. Comparison techniques for difference equations are used to find sufficient conditions for population persistence or extinction. The persistence and extinction theorem is used to define the critical patch number, the threshold for population persistence. Several examples are presented which illustrate the results of the theorems. The model is applied to a watersnake population.     

THE IMPACT OF AGE-, SPACE-AND STOCHASTIC-STRUCTURE ON THE EXTINCTION PROBABILITY OF A CHINOOK SALMON METAPOPULATION

ABSTRACT. Using a mechanistic model, based on chinook life history, incorporating environmental and demographic stochasticity, we investigate how the probability of extinction is controlled by age, space and stochastic structure. Environmental perturbations of age dependent survivorships, combined with mixing of year classes in the spawning population, can lower the probability of extinction dramatically. This is an analog of the more familiar metapopulation result where dispersal between asynchronously fluctuating populations enhances persistence. For a two-river chinook metapopulation, dispersal between rivers with asynchronous environmental perturbations also dramatically enhances persistence, and anti-synchronous population fluctuations provide an even greater persistence probability. Anti-synchronous fluctuations would most likely occur in pristine habitat with naturally high levels of heterogeneity. Fifty percent dispersal between two populations provides the greatest insurance against extinction, a rate unrealistically high for salmon. In contrast, dispersal between exactly correlated populations with large amplitude environmental perturbations does not help persistence, no matter how high the dispersal rate. This is in spite of weak asynchrony provided by demographic stochasticity. Dispersal between rivers, one degraded and the other pristine, can substantially increase the probability of metapopulation extinction. Population structure, combined with asynchronous environmental perturbations and dispersal (or age class mixing) lowers the probability of chinook extinction dramatically but is almost useless when survivorships are impaired.     

Propagation of second order integrodifference equations with local monotonicity

This paper is concerned with the spreading speeds and traveling wavefronts of second order integrodifference equations with local monotonicity. By introducing two auxiliary integrodifference equations, the spreading speed and traveling wave solutions are studied. In particular, we obtain the nonexistence of monotone traveling wave solutions for an example if it is local monotone. These results are applied to a model which is obtained by introducing the spatial variable to a difference equation used by the International Whaling Commission.     

Spreading speeds and traveling wavefronts for second order integrodifference equations

This paper is concerned with the spreading speeds and traveling wavefronts for second order integrodifference equations. By introducing an auxiliary integrodifference system, the spreading speed is established for the integrodifference equation. It is shown that the spreading speed coincides with the minimal wave speed for monotonic traveling wavefronts. Furthermore, we prove that the traveling wavefronts are stable by applying the squeezing technique. Finally, we analyze the different effects of the delay term appearing in the integrodifference equation from the viewpoint of ecology.     

Discrete-time Metapopulation Dynamics and Unidirectional Dispersal

The effects of unidirectional dispersal on single pioneer species discrete-time metapopulations where the pre-dispersal local patch dynamics are of the same (compensatory or overcompensatory) or mixed (compensatory and overcompensatory) types are studied. Single-species unidirectional metapopulation models behave as single-species single-patch models whenever all pre-dispersal local patch dynamics are compensatory and the dispersal rate is low. The pioneer species goes extinct in at least one patch when the dispersal rate is high, while it persists when the rate is low. Unidirectional dispersal can generate multiple attractors with fractal basin boundaries whenever the pre-dispersal local patch dynamics are overcompensatory, and is capable of altering the local patch dynamics in mixed systems from compensatory to overcompensatory dynamics and vice versa.     

HOW BIG AND HOW CLOSE? HABITAT PATCH SIZE AND SPACING TO CONSERVE A THREATENED SPECIES

Abstract. We present results of a spatially explicit, individual‐based stochastic dispersal model (HexSim) to evaluate effects of size and spacing of patches of habitat of Northern Spotted Owls (NSO; Strix occidentalis caurina) in Pacific Northwest, USA, to help advise recovery planning efforts. We modeled 31 artificial landscape scenarios representing combinations of NSO habitat cluster size (range 4–49 NSO pairs per cluster) and edge‐to‐edge cluster spacing (range 7–101 km), and an all‐habitat landscape. We ran scenarios using empirical estimates of NSO dispersal dynamics and distances and stage class vital rates (representing current population declines) and under adult survival rates adjusted to achieve an initially stationary population. Results suggested that long‐term (100‐yr) habitat occupancy rates are significantly higher with habitat clusters supporting ≥25 NSO pairs and ≤15 km spacing, and with overall landscapes of ≥35–40% habitat. Although habitat provision is key to NSO recovery, no habitat configuration provided for long‐term population persistence when coupled with currently observed vital rates. Results also suggested a key role of floaters (unpaired, nonterritorial, dispersing owls) in recolonizing vacant habitat, and that the floater population segment becomes increasingly depleted with greater population declines. We suggest additional areas of modeling research on this and other threatened species.     

Effects of population dispersal and impulsive control tactics on pest management

In this paper, we firstly consider a Lotka–Volterra predator–prey model with impulsive constant releasing for natural enemies and a proportion of killing or catching pests at fixed moments, and we have proved that there exists a pest-eradication periodic solution which is globally asymptotically stable. Further, we extend the model for the population to move in a two-patch environment. The effects of population dispersal and impulsive control tactics are investigated, i.e. we chiefly address the question of whether population dispersal is beneficial or detrimental for pest persistence. To do this, some special cases are theoretically investigated and numerical investigations are done for general case. The results indicate that for some ranges of dispersal rates, population dispersal is beneficial to pest control, but for other ranges, it is harmful. These clarify that we can get some new effective pest control strategies by controlling the dispersal rates of pests and natural enemies.     

DYNAMICS OF AN AGE‐STRUCTURED METAPOPULATION MODEL

ABSTRACT. We introduce a metapopulation model that includes both landscape changes (patch destruction and recreation) and age‐dependent metapopulation dynamics. A threshold quantity is derived and related to the existence of an ecologically nontrivial equilibrium, to the stability of the species‐free equilibrium, and to weak and strong persistence of the species. We provide examples to illustrate how age‐related changes in patch colonization and extinction rates can alter metapopulation persistence. Future field studies may need to address the temporal dynamics that characterize local populations in fragmented landscapes.     

一类捕食与被捕食LV模型的扩散性质

本文证明了一类带有扩散的捕食与被捕食Ｌｏｔｋａ－Ｖｏｌｔｅｒｒａ模型的如下性质：当该模型存在正平衡点时,它的一切正解是强持续生存的;当扩散率较小时,该系统的正平衡点是稳定的;当扩散率增大且位于某一开区间内变化时,该系统的正平衡点是不稳定的,而且分支出唯一的小振幅空间周期解;当扩散率继续增大时,该系统的正平衡点又变为稳定的．     

脆弱斑块中植物种子的脉冲扩散

考虑了一类脆弱斑块中植物种子的脉冲漂移模型,得到了系统永久持续生存性.在此基础上,利用单调凸算子理论,得到了系统唯一全局渐进稳定的周期解.数值模拟也表明脉冲扩散能挽救脆弱斑块中的植物种群.     

Permanence of a single-species dispersal system and predator survival

This paper considers permanence of a single-species dispersal periodic system with the possibility of the loss for the species during their dispersion among patches. The condition obtained for permanence generalizes the known condition on the system without loss for the species in the process of movement. Next, we add predators into every patch and consider the survival possibility of the predator. It is shown that the total amount of the predators can remain positive, if the single-species (prey) dispersal system has a positive periodic solution and the quantity of prey in each patch is enough for survival of the predator.