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.     

Dynamics of a delayed Lotka–Volterra competition model with directed dispersal

We consider a diffusive Lotka–Volterra competition system with stage structure, where the intrinsic growth rates and the carrying capacities of the species are assumed spatially heterogeneous. Here, we also assume each of the competing populations chooses its dispersal strategy as the tendency to have a distribution proportional to a certain positive prescribed function. We give the effects of dispersal strategy, delay, the intrinsic growth rates and the competition parameters on the global dynamics of the delayed reaction diffusion model. Our result shows that competitive exclusion occurs when one of the diffusion strategies is proportional to the carrying capacity, while the other is not; while both populations can coexist if the competition favors the latter species. Finally, we point out that the method is also applied to the global dynamics of other kinds of delayed competition models.     

Global dynamics of a competition model with non-local dispersal I: The shadow system

Equations with non-local dispersal have been widely used as models in biology. In this paper we focus on logistic models with non-local dispersal, for both single and two competing species. We show the global convergence of the unique positive steady state for the single equation and derive various properties of the positive steady state associated with the dispersal rate. We investigate the effects of dispersal rates and inter-specific competition coefficients in a shadow system for a two-species competition model and completely determine the global dynamics of the system. Our results illustrate that the effect of dispersal in spatially heterogeneous environments can be quite different from that in homogeneous environments.     

Evolution in spatial predator–prey models and the “prudent predator”: The inadequacy of steady‐state organism fitness and the concept of individual and group selection

We review recent research which reveals: (1) how spatially distributed populations avoid overexploiting resources due to the local extinction of over‐exploitative variants, and (2) how the conventional understanding of evolutionary processes is violated by spatial populations so that basic concepts, including fitness assignment to individual organisms, are not applicable, and even kin and group selection are unable to describe the mechanism by which exploitative behavior is bounded. To understand these evolutionary processes, a broader view is needed of the properties of multiscale spatiotemporal patterns in organism–environment interactions. We discuss measures that quantify the effects of these interactions on the evolution of a population, including multigenerational fitness and the heritability of the environment. © 2008 Wiley Periodicals, Inc. Complexity, 2008.     

Inhomogeneous maps and mathematical theory of selection

In this paper we develop a theory of general selection systems with discrete time and explore the evolution of selection systems, in particular, inhomogeneous populations. We show that the knowledge of the initial distribution of the selection system allows us to determine explicitly the system distribution at the entire time interval. All statistical characteristics of interest, such as mean values of the fitness or any trait can be predicted effectively for indefinite time and these predictions dramatically depend on the initial distribution. The Fisher Fundamental theorem of natural selection (FTNS) and more general the Price equations are the famous results of the mathematical selection theory. We show that the problem of dynamic insufficiency for the Price equations and for the FTNS can be resolved within the framework of selection systems. Effective formulas for solutions of the Price equations and for the FTNS are derived. Applications of the developed theory to some other problems of mathematical biology (dynamics of inhomogeneous logistic and Ricker model, selection in rotifer populations) are also given. Complex behavior of the total population size, the mean fitness (in contrast to the plain FTNS) and other traits is possible for inhomogeneous populations with density-dependent fitness. The temporary dynamics of these quantities can be investigated with the help of suggested methods.     

Does movement toward better environments always benefit a population?

We study the effects of advection along environmental gradients on logistic reaction-diffusion models for population growth. The local population growth rate is assumed to be spatially inhomogeneous, and the advection is taken to be a multiple of the gradient of the local population growth rate. It is also assumed that the boundary acts as a reflecting barrier to the population. We show that the effects of such advection depend crucially on the shape of the habitat of the population: if the habitat is convex, the movement in the direction of the gradient of the growth rate is always beneficial to the population, while such advection could be harmful for certain non-convex habitats.     

Aggregation and heterogeneity from the nonlinear dynamic interaction of birth,maturation and spatial migration

We consider a single species structured population distributed in two identical patches connected by spatial dispersal. Assuming that the maturation time for each individual is a random variable with a gamma distribution and that the spatial dispersal rate is constant, we obtain from a hyperbolic differential equation a system of six ordinary differential equations for the matured populations and their moments. Our qualitative analysis and numerical simulations show that the nonlinear interaction of birth process, the maturation delay and the spatial dispersal can lead to a new mechanism for individual aggregation in the form of the existence of multiple stable heterogeneous equilibria, even though the spatial dispersal is assumed to be proportional to the population gradients with a constant rate.     

ECONOMICALLY OPTIMAL MARINE RESERVES WITHOUT SPATIAL HETEROGENEITY IN A SIMPLE TWO‐PATCH MODEL

Bioeconomic analyses of spatial fishery models have established that marine reserves can be economically optimal (i.e., maximize sustainable profit) when there is some type of spatial heterogeneity in the system. Analyses of spatially continuous models and models with more than two discrete patches have also demonstrated that marine reserves can be economically optimal even when the system is spatially homogeneous. In this note we analyze a spatially homogeneous two‐patch model and show that marine reserves can be economically optimal in this case as well. The model we study includes the possibility that fishing can damage habitat. In this model, marine reserves are necessary to maximize sustainable profit when dispersal between the patches is sufficiently high and habitat is especially vulnerable to damage.     

On stochastic estimation

We consider a local random searching method to approximate a root of a specified equation. If such roots, which can be regarded as estimators for the Euclidean parameter of a statistical experiment, have some asymptotic optimality properties, the local random searching method leads to asymptotically optimal estimators in such cases. Application to simple first order autoregressive processes and some simulation results for such models are also included.     

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.     

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.     

POPULATION EXTINCTION IN DETERMINISTICAND STOCHASTIC DISCRETE‐TIME EPIDEMIC MODELS WITH PERIODIC COEFFICIENTS WITH APPLICATIONS TO AMPHIBIAN POPULATIONS

ABSTRACT. Discrete‐time deterministic and stochastic epidemic models are formulated for the spread of disease in a structured host population. The models have applications to a fungal pathogen affecting amphibian populations. The host population is structured according to two developmental stages, juveniles and adults. The juvenile stage is a post‐metamorphic, nonreproductive stage, whereas the adult stage is reproductive. Each developmental stage is further subdivided according to disease status, either susceptible or infected. There is no recovery from disease. Each year is divided into a fixed number of periods, the first period represents a time of births and the remaining time periods there are no births, only survival within a stage, transition to another stage or transmission of infection. Conditions are derived for population extinction and for local stability of the disease‐free equilibrium and the endemic equilibrium. It is shown that high transmission rates can destabilize the disease‐free equilibrium and low survival probabilities can lead to population extinction. Numerical simulations illustrate the dynamics of the deterministic and stochastic models.     

INDIVIDUAL‐BASED MODELS AND THE MANAGEMENT OF SHOREBIRD POPULATIONS

Abstract Individual‐based models (IBMs) predict how animal populations will be affected by changes in their environment by modeling the responses of fitness‐maximizing individuals to environmental change and by calculating how their aggregate responses change the average fitness of individuals and thus the demographic rates, and therefore size of the population. This paper describes how the need to develop a new approach to make such predictions was identified in the mid‐1970s following work done to predict the effect of building a freshwater reservoir on part of the intertidal feeding areas of the shorebirds Charadrii that overwinter on the Wash, a large embayment on the east coast of England. The paper describes how the approach was developed and tested over 20 years (1976–1995) on a population of European oystercatchers Haematopus ostralegus eating mussels Mytilus edulis on the Exe estuary in Devon, England. The paper goes on to describe how individual‐based modeling has been applied over the last 10 years to a wide range of environmental issues and to many species of shorebirds and wildfowl in a number of European countries. Although it took 20 years to develop the approach for 1 bird species on 1 estuary, ways have been found by which it can now be applied quite rapidly to a wide range of species, at spatial scales ranging from 1 estuary to the whole continent of Europe. This can now be done within the time period typically allotted to environmental impact assessments involving coastal bird populations in Europe. The models are being used routinely to predict the impact on the fitness of coastal shorebirds and wildfowl of habitat loss from (i) development, such as building a port over intertidal flats; (ii) disturbance from people, raptors, and aircraft; (iii) harvesting shellfish; and (iv) climate change and any associated rise in sea level. The model has also been used to evaluate the probable effectiveness of mitigation measures aimed at ameliorating the impact of such environmental changes on the birds. The first steps are now being taken to extend the approach to diving sea ducks and farmland birds during the nonbreeding season. The models have been successful in predicting the observed behavior and mortality rates in winter of shorebirds on a number of European estuaries, and some of the most important of these tests are described. These successful tests of model predictions raise confidence that the model can be used to advise policy makers concerned with the management of the coast and its important bird populations.     

ANALYSIS OF SENSITIVITY AND UNCERTAINTY IN AN INDIVIDUAL‐BASED MODEL OF A THREATENED WILDLIFE SPECIES

Sensitivity analysis—determination of how prediction variables affect response variables—of individual‐based models (IBMs) are few but important to the interpretation of model output. We present sensitivity analysis of a spatially explicit IBM (HexSim) of a threatened species, the Northern Spotted Owl (NSO; Strix occidentalis caurina) in Washington, USA. We explored sensitivity to HexSim variables representing habitat quality, movement, dispersal, and model architecture; previous NSO studies have well established sensitivity of model output to vital rate variation. We developed “normative” (expected) model settings from field studies, and then varied the values of ≥ 1 input parameter at a time by ±10% and ±50% of their normative values to determine influence on response variables of population size and trend. We determined time to population equilibration and dynamics of populations above and below carrying capacity. Recovery time from small population size to carrying capacity greatly exceeded decay time from an overpopulated condition, suggesting lag time required to repopulate newly available habitat. Response variables were most sensitive to input parameters of habitat quality which are well‐studied for this species and controllable by management. HexSim thus seems useful for evaluating potential NSO population responses to landscape patterns for which good empirical information is available.     

Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats

The current paper is devoted to the study of spatial spreading dynamics of monostable equations with nonlocal dispersal in spatially periodic habitats. In particular, the existence and characterization of spreading speeds is considered. First, a principal eigenvalue theory for nonlocal dispersal operators with space periodic dependence is developed, which plays an important role in the study of spreading speeds of nonlocal periodic monostable equations and is also of independent interest. In terms of the principal eigenvalue theory it is then shown that the monostable equation with nonlocal dispersal has a spreading speed in every direction in the following cases: the nonlocal dispersal is nearly local; the periodic habitat is nearly globally homogeneous or it is nearly homogeneous in a region where it is most conducive to population growth in the zero-limit population. Moreover, a variational principle for the spreading speeds is established.     

Modelling and stability analysis in human population genetics with selection and mutation

Population genetics is a scientific discipline that has extensively benefitted from mathematical modelling; since the Hardy‐Weinberg law (1908) to date, many mathematical models have been designed to describe the genotype frequencies evolution in a population. Existing models differ in adopted hypothesis on evolutionary forces (such as, for example, mutation, selection, and migration) acting in the population. Mathematical analysis of population genetics models help to understand if the genetic population admits an equilibrium, ie, genotype frequencies that will not change over time. Nevertheless, the existence of an equilibrium is only an aspect of a more complex issue concerning the conditions that would allow or prevent populations to reach the equilibrium. This latter matter, much more complex, has been only partially investigated in population genetics studies. We here propose a new mathematical model to analyse the genotype frequencies distribution in a population over time and under two major evolutionary forces, namely, mutation and selection; the model allows for both infinite and finite populations. In this paper, we present our model and we analyse its convergence properties to the equilibrium genotype frequency; we also derive conditions allowing convergence. Moreover, we show that our model is a generalisation of the Hardy‐Weinberg law and of subsequent models that allow for selection or mutation. Some examples of applications are reported at the end of the paper, and the code that simulates our model is available online at https://www.ding.unisannio.it/persone/docenti/del-vecchio for free use and testing.     

函数优化的小生境蝙蝠算法

针对蝙蝠算法易陷入局部最优解的缺点,利用小生境技术对蝙蝠算法进行了改进,提出一种小生境蝙蝠优化算法.算法基于小生境技术的适应度共享来分隔种群,引入了小生境排挤机制来保持种群多样性,在延续蝙蝠算法原有并行搜索等优势的基础上,提高了算法的金局搜索能力和局部收敛速度,具有可在不同邻域内发现多个解的特点.通过对一系列经典函数测试,并与已有算法进行比较,结果表明该算法在函数优化问题的求解中具有较高的计算效率和精度,以及较好的全局寻优能力.     

The Markov structure of population growth

Some problems concerning the development of general models of population growth are examined, with particular reference to Markovian modelling. The tenet of the paper is that such models can be more realistically attempted by focusing on temporal rather than spatial modifications. Specifically, the time concept considered is linked to the dependent structure inherent in the partial order of descent from mother to child. The author attempts to develop "a general theory of populations of individuals under what might be called free reproduction. Hence, the only dependence assumed between individuals is that from mothers to children." He suggests that the results "can be translated into assertions about evolution in real, physical time and also about the final, stable or balanced, composition of populations, over ages, types, family structure, and many other aspects of populations."     

Populations with individual variation in dispersal in heterogeneous environments: Dynamics and competition with simply diffusing populations

We consider a model for a population in a heterogeneous environment, with logistic-type local population dynamics, under the assumption that individuals can switch between two different nonzero rates of diffusion. Such switching behavior has been observed in some natural systems. We study how environmental heterogeneity and the rates of switching and diffusion affect the persistence of the population. The reactiondiffusion systems in the models can be cooperative at some population densities and competitive at others. The results extend our previous work on similar models in homogeneous environments. We also consider competition between two populations that are ecologically identical, but where one population diffuses at a fixed rate and the other switches between two different diffusion rates. The motivation for that is to gain insight into when switching might be advantageous versus diffusing at a fixed rate. This is a variation on the classical results for ecologically identical competitors with differing fixed diffusion rates, where it is well known that "the slower diffuser wins".     

Baseline models of spatial population dynamics

To understand human population dynamics fully, before considering complex human agency it may be useful to construct baseline models to see where such agency may and may not be necessary. In fact, the dynamics of human populations may be amenable to mathematical modeling with relatively parsimonious mechanisms. We review some of the more prominent of such models, namely, the spatial Galton-Watson (GW) model, modifications of the GW model that add migration and immigration, and the Bolker-Pacala model, in which mortality (or birth rate) is affected by competition. We show that change in the distribution of population density over the last century for 12 American rural states may be captured by the simplest of the models, the spatial GW model.