Search

In this paper, we use an integrodifference equation model and pairwise invasion analysis to find what dispersal strategies are evolutionarily stable strategies (also known as evolutionarily steady or ESS) when there is spatial heterogeneity and possibly seasonal variation in habitat suitability. In that case there are both advantages and disadvantages of dispersing. We begin with the case where all spatial locations can support a viable population, and then consider the case where there are non-viable regions in the habitat. If the viable regions vary seasonally, and the viable regions in summer and winter do not overlap, dispersal may really be necessary for sustaining a population. Our findings generally align with previous findings in the literature that were based on other modeling frameworks, namely that dispersal strategies associated with ideal free distributions are evolutionarily stable. In the case where only part of the habitat can sustain a population, we show that a partial occupation ideal free distribution that occupies only the viable region is associated with a dispersal strategy that is evolutionarily stable. As in some previous works, the proofs of these results make use of properties of line sum symmetric functions, which are analogous to those of line sum symmetric matrices but applied to integral operators.

For any $N \geq 2$, we show that there are choices of diffusion rates $\{d_i\}_{i=1}^N$ such that for $N$ competing species which are ecologically identical and have distinct diffusion rates, the s...

We analyze a reaction-diffusion system modeling the competition of multiple phytoplankton species which are limited only by light. While the dynamics of a single species has been well studied, the dynamics of the two-species model has only begun to be understood with the recent establishment of a comparison principle. In this paper, we show that the competition of \begin{document}$ N $\end{document} similar phytoplankton species, for any number \begin{document}$ N $\end{document} , generically leads to competitive exclusion. The main tool is the theory of a normalized principal bundle for linear parabolic equations.

A population is said to have an ideal free distribution in a spatially heterogeneous but temporally constant environment if each of its members have chosen a fixed spatial location in a way that optimizes its individual fitness, allowing for the effects of crowding. In this paper, we extend the idea of individual fitness associated with a specific location in space to account for the full path that an individual organism takes in space and time over a periodic cycle, and extend the mathematical formulation of an ideal free distribution to general time periodic environments. We find that, as in many other cases, populations using dispersal strategies that can produce a generalized ideal free distribution have a competitive advantage relative to populations using strategies that do not produce an ideal free distribution. A sharp criterion on the environmental functions is found to be necessary and sufficient for such ideal free distribution to be feasible. In the case the criterion is met, we showed that there exist dispersal strategies that can be identified as producing a time-periodic version of an ideal free distribution, and such strategies are evolutionarily steady and are neighborhood invaders from the viewpoint of adaptive dynamics. Our results extend previous works in which the environments are either temporally constant, or temporally periodic but the total carrying capacity is temporally constant.

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 reaction diffusion 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., To be published in SCIENCE CHINA Mathematics

Consumers must track and acquire resources in complex landscapes. Much discussion has focused on the concept of a ‘resource gradient’ and the mechanisms by which consumers can take advantage of such gradients as they navigate their landscapes in search of resources. However, the concept of tracking resource gradients means different things in different contexts. Here we take a synthetic approach and consider six different definitions of what it means to search for resources based on density or gradients in density. These include scenarios where consumers change their movement behavior based on the density of conspecifics, on the density of resources, and on spatial or temporal gradients in resources. We also consider scenarios involving non-local perception and a form of memory. Using a continuous space, continuous time model that allows consumers to switch between resource-tracking and random motion, we investigate the relative performance of these six different strategies. Consumers’ success in matching the spatiotemporal distributions of their resources differs starkly across the six scenarios. Movement strategies based on perception and response to temporal (rather than spatial) resource gradients afforded consumers with the best opportunities to match resource distributions. All scenarios would allow for optimization of resource matching in terms of the underlying parameters, providing opportunities for evolutionary adaptation, and links back to classical studies of foraging ecology.

Seasonal migrations are a widespread and broadly successful strategy for animals to exploit periodic and localized resources over large spatial scales. It remains an open and largely case-specific question whether long-distance migrations are resilient to environmental disruptions. High levels of mobility suggest an ability to shift ranges that can confer resilience. On the other hand, a conservative, hard-wired commitment to a risky behavior can be costly if conditions change. Mechanisms that contribute to migration include identification and responsiveness to resources, sociality, and cognitive processes such as spatial memory and learning. Our goal was to explore the extent to which these factors interact not only to maintain a migratory behavior but also to provide resilience against environmental changes. We develop a diffusion-advection model of animal movement in which an endogenous migratory behavior is modified by recent experiences via a memory process, and animals have a social swarming-like behavior over a range of spatial scales. We found that this relatively simple framework was able to adapt to a stable, seasonal resource dynamic under a broad range of parameter values. Furthermore, the model was able to acquire an adaptive migration behavior with time. However, the resilience of the process depended on all the parameters under consideration, with many complex trade-offs. For example, the spatial scale of sociality needed to be large enough to capture changes in the resource, but not so large that the acquired collective information was overly diluted. A long-term reference memory was important for hedging against a highly stochastic process, but a higher weighting of more recent memory was needed for adapting to directional changes in resource phenology. Our model provides a general and versatile framework for exploring the interaction of memory, movement, social and resource dynamics, even as environmental conditions globally are undergoing rapid change.

Animals use different modes of movement at different times, in different locations, and on different scales. Incorporating such context dependence in mathematical models represents a substantial increase in complexity, but creates an opportunity to more fully integrate key biological features. Here, we consider the spatial dynamics of a population of foragers with two subunits. In one subunit, foragers move via diffusion (random search) whereas in the other, foragers move via advection (gradient-following search). Foragers switch back and forth between the subunits as functions of their spatial context (i.e., depending on whether they are inside or outside of a patch, or depending on whether or not they can detect a gradient in resource density). We consider a one-dimensional binary landscape of resource patches and non-habitat and gauge success in terms of how well the mobile foragers overlap with the distribution of resources. Actively switching between dispersal modes can sometimes greatly enhance this spatial overlap relative to the spatial overlap possible when foragers merely blend advection and diffusion modes at all times. Switching between movement modes is most beneficial when organism’s gradient-following abilities are weak compared to its overall capacity for movement, but switching can actually be quite detrimental for organisms that can rapidly follow resource gradients. An organism’s perceptual range plays a critical role in determining the conditions under which switching movement modes benefits versus disadvantages foragers as they seek out resources.

Although dispersal is generally viewed as a crucial determinant for the fitness of any organism, our understanding of its role in the persistence and spread of plant populations remains incomplete. Generalizing and predicting dispersal processes are challenging due to context dependence of seed dispersal, environmental heterogeneity and interdependent processes occurring over multiple spatial and temporal scales. Current population models often use simple phenomenological descriptions of dispersal processes, limiting their ability to examine the role of population persistence and spread, especially under global change. To move seed dispersal ecology forward, we need to evaluate the impact of any single seed dispersal event within the full spatial and temporal context of a plant’s life history and environmental variability that ultimately influences a population’s ability to persist and spread. In this perspective, we provide guidance on integrating empirical and theoretical approaches that account for the context dependency of seed dispersal to improve our ability to generalize and predict the consequences of dispersal, and its anthropogenic alteration, across systems. We synthesize suitable theoretical frameworks for this work and discuss concepts, approaches and available data from diverse subdisciplines to help operationalize concepts, highlight recent breakthroughs across research areas and discuss ongoing challenges and open questions. We address knowledge gaps in the movement ecology of seeds and the integration of dispersal and demography that could benefit from such a synthesis. With an interdisciplinary perspective, we will be able to better understand how global change will impact seed dispersal processes, and potential cascading effects on plant population persistence, spread and biodiversity., Although dispersal is generally viewed as a crucial determinant for the fitness of any organism, our understanding of its role in the persistence and spread of plant populations remains incomplete. In this perspective, we provide guidance on integrating empirical and theoretical approaches that account for the context dependency of seed dispersal to improve our ability to generalize and predict the consequences of dispersal, and its anthropogenic alteration, across systems. With an interdisciplinary perspective, we will be able to better understand how global change will impact seed dispersal processes, and potential cascading effects on plant population persistence, spread and biodiversity.

Roughly speaking, a population is said to have an ideal free distribution on a spatial region if all of its members can and do locate themselves in a way that optimizes their fitness, allowing for the effects of crowding. Dispersal strategies that can lead to ideal free distributions of populations using them have been shown to exist and to be evolutionarily stable in a number of modeling contexts in the case of habitats that vary in space but not in time. Those modeling contexts include reaction-diffusion-advection models and the analogous models using discrete diffusion or nonlocal dispersal described by integro-differential equations. Furthermore, in the case of reaction-diffusion-advection models and their nonlocal analogues, there are strategies that allow populations to achieve an ideal free distribution by using only local information about environmental quality and/or gradients. We show that in the context of reaction-diffusion-advection models for time-periodic environments with spatially varying resource levels, where the total level of resources in an environment remains fixed but its location varies seasonally, there are strategies that allow populations to achieve an ideal free distribution. We also show that those strategies are evolutionarily stable. However, achieving an ideal free distribution in a time-periodic environment requires the use of nonlocal information about the environment such as might be derived from experience and memory, social learning, or genetic programming.

Chikungunya, dengue, and Zika viruses are all transmitted by Aedes aegypti and Aedes albopictus mosquito species, had been imported to Florida and caused local outbreaks. We propose a deterministic model to study the importation and local transmission of these mosquito-borne diseases. The purpose is to model and mimic the importation of these viruses to Florida via travelers, local infections in domestic mosquitoes by imported travelers, and finally non-travel related transmissions to local humans by infected local mosquitoes. As a case study, the model will be used to simulate the accumulative Zika cases in Florida. Since the disease system is driven by a continuing input of infections from outside sources, orthodox analytic methods based on the calculation of the basic reproduction number are inadequate to describe and predict their behavior. Via steady-state analysis and sensitivity analysis, effective control and prevention measures for these mosquito-borne diseases are tested.

In this article we study the long term behavior of the competitive system { ∂ u ∂ t = ∇ ⋅ [ α ( x ) ∇ u m ] + u ( m ( x ) − u − b v ) in Ω , t > 0 , ∂ u ∂ t = ∇ ⋅ [ β ( x ) ∇ v ] + v ( m ( x ) − c u − v ) in Ω , t > 0 , ∇ u m ⋅ n ˆ = ∇ v ⋅ n ˆ = 0 on ∂ Ω , t > 0 , which supports for the first species an ideal free distribution, that is a positive steady state which matches the per-capita growth rate. Previous results have stated that when b = c = 1 the ideal free distribution is an evolutionarily stable and neighborhood invader strategy, that is the species with density v always goes extinct. Thus, of particular interest will be to study the interplay between the inter-specific competition coefficients b , c and the diffusion coefficients α ( x ) and β ( x ) on the critical values for stability of semi-trivial steady states, and the structure of bifurcation branches of positive equilibria arising from these equilibria. We will also show that under certain regimes the system sustains multiple positive steady states.

Most classical models for the movement of organisms assume that all individuals have the same patterns and rates of movement (for example, diffusion with a fixed diffusion coefficient) but there is empirical evidence that movement rates and patterns may vary among different individuals. A simple way to capture variation in dispersal that has been suggested in the ecological literature is to allow individuals to switch between two distinct dispersal modes. We study models for populations whose members can switch between two different nonzero rates of diffusion and whose local population dynamics are subject to density dependence of logistic type. The resulting models are reaction-diffusion systems that can be cooperative at some population densities and competitive at others. We assume that the focal population inhabits a bounded region and study how its overall dynamics depend on the parameters describing switching rates and local population dynamics. (Traveling waves and spread rates have been studied for similar models in the context of biological invasions.) The analytic methods include ideas and results from reaction-diffusion theory, semi-dynamical systems, and bifurcation/continuation theory.

Motivated by evolutionary biology, we study general infinite-dimensional dynamical systems involving two species – the resident and the invader. Sufficient conditions for competition exclusion phenomena are given when the two species play similar, but distinct, strategies. Those conditions are based on invasibility criteria, for instance, evolutionarily stable strategies in the framework of adaptive dynamics. These types of questions were first proposed and studied by S. Geritz et al. [20] and S. Geritz [19] for a class of ordinary differential equations. We extend and generalize previous work in two directions. Firstly, we consider analytic semiflows in infinite-dimensional spaces. Secondly, we devise an argument based on Hadamard's graph transform method that does not depend on the monotonicity of the two-species system. Our results are applicable to a wide class of reaction–diffusion models as well as models with nonlocal diffusion operators.

In this paper we employ a discrete-diffusion modeling framework to examine a system inspired by the nano-ecology experiments on the bacterium Escherichia coli reported upon in Keymer et al. (2006). In these experiments, the bacteria inhabit a linear array of 85 ``microhabitat patches (MHP's)", linked by comparatively thinner corridors through which bacteria may pass between adjacent MHP's. Each MHP is connected to its own source of nutrient substrate, which flows into the MHP at a rate that can be controlled in the experiment. Logistic dynamics are assumed within each MHP, and nutrient substrate flow determines the prediction of the within MHP dynamics in the absence of bacteria dispersal between patches. Patches where the substrate flow rate is sufficiently high sustain the bacteria in the absence of between patch movement and may be regarded as sources, while those with insufficient substrate flow lead to the extinction of the bacteria in the within patch environment and may be regarded as sinks. We examine the role of dispersal in determining the predictions of the model under source-sink dynamics.

Seed dispersal enables plants to reach hospitable germination sites and escape natural enemies. Understanding when and how much seed dispersal matters to plant fitness is critical for understanding plant population and community dynamics. At the same time, the complexity of factors that determine if a seed will be successfully dispersed and subsequently develop into a reproductive plant is daunting. Quantifying all factors that may influence seed dispersal effectiveness for any potential seed-vector relationship would require an unrealistically large amount of time, materials and financial resources. On the other hand, being able to make dispersal predictions is critical for predicting whether single species and entire ecosystems will be resilient to global change. Building on current frameworks, we here posit that seed dispersal ecology should adopt plant functional groups as analytical units to reduce this complexity to manageable levels. Functional groups can be used to distinguish, for their constituent species, whether it matters (i) if seeds are dispersed, (ii) into what context they are dispersed and (iii) what vectors disperse them. To avoid overgeneralization, we propose that the utility of these functional groups may be assessed by generating predictions based on the groups and then testing those predictions against species-specific data. We suggest that data collection and analysis can then be guided by robust functional group definitions. Generalizing across similar species in this way could help us to better understand the population and community dynamics of plants and tackle the complexity of seed dispersal as well as its disruption., Seed dispersal is critical to plant fitness and plant community dynamics. However, measuring and tracking all factors that influence seed dispersal effectiveness for any potential seed-disperser relationship would require an unrealistically large amount of time, materials and financial resources. Building on current frameworks, we suggest that seed dispersal ecology quantify seed dispersal at the scale of plant functional groups, in order to reduce complexity to manageable levels. Based on functional group classifications, plant species can be distinguished by their level of dependence on seed dispersal and its mechanisms.

West Nile virus (WNV) was first detected in the United States (U.S.) during an outbreak in New York City in 1999 with 62 human cases including seven deaths. In 2001, the first human case in Florida was identified, and in Texas and California it was 2002 and 2004, respectively. WNV has now been spread to almost all states in the US. In 2015, the Center for Disease Control and Prevention (CDC) reported 2,175 human cases, including 146 deaths, from 45 states. WNV is maintained in a cycle between mosquitoes and animal hosts in which birds are the predominant and preferred reservoirs while most mammals, including humans, are considered dead-end hosts, as they do not appear to develop high enough titers of WNV in the blood to infect mosquitoes. In this article, we propose a deterministic model by including interactions among mosquitoes, birds, and humans to study the local transmission dynamics of WNV. To validate the model, it is used to simulate the WNV human data of infected cases and accumulative deaths from 1999 to 2013 in the states of New York, Florida, Texas, and California as reported to the CDC. These simulations demonstrate that the epidemic of WNV in New York, Texas, and California (and thus in the U.S.) has not reached its equilibrium yet and may be expected to get worse if the current control strategies are not enhanced. Mathematical and numerical analyses of the model are carried out to understand the transmission dynamics of WNV and explore effective control measures for the local outbreaks of the disease. Our studies suggest that the larval mosquito control measure should be taken as early as possible in a season to control the mosquito population size and the adult mosquito control measure is necessary to prevent the transmission of WNV from mosquitoes to birds and humans.

In this article, we study how the rates of diffusion in a reaction-diffusion model for a stage structured population in a heterogeneous environment affect the model&rsquo, s predictions of persistence or extinction for the population. In the case of a population without stage structure, faster diffusion is typically detrimental. In contrast to that, we find that, in a stage structured population, it can be either detrimental or helpful. If the regions where adults can reproduce are the same as those where juveniles can mature, typically slower diffusion will be favored, but if those regions are separated, then faster diffusion may be favored. Our analysis consists primarily of estimates of principal eigenvalues of the linearized system around ( 0 , 0 ) and results on their asymptotic behavior for large or small diffusion rates. The model we study is not in general a cooperative system, but if adults only compete with other adults and juveniles with other juveniles, then it is. In that case, the general theory of cooperative systems implies that, when the model predicts persistence, it has a unique positive equilibrium. We derive some results on the asymptotic behavior of the positive equilibrium for small diffusion and for large adult reproductive rates in that case.

When climatic or environmental conditions change, plant populations must either adapt to these new conditions, or track their niche via seed dispersal. Adaptation of plants to different abiotic environments has mostly been discussed with respect to physiological and demographic parameters that allow local persistence. However, rapid modifications in response to changing environmental conditions can also affect seed dispersal, both via plant traits and via their dispersal agents. Studying such changes empirically is challenging, due to the high variability in dispersal success, resulting from environmental heterogeneity, and substantial phenotypic variability of dispersal-related traits of seeds and their dispersers. The exact mechanisms that drive rapid changes are often not well understood, but the ecological implications of these processes are essential determinants of dispersal success, and deserve more attention from ecologists, especially in the context of adaptation to global change. We outline the evidence for rapid changes in seed dispersal traits by discussing variability due to plasticity or genetics broadly, and describe the specific traits and biological systems in which variability in dispersal is being studied, before discussing some of the potential underlying mechanisms. We then address future research needs and propose a simulation model that incorporates phenotypic plasticity in seed dispersal. We close with a call to action and encourage ecologists and biologist to embrace the challenge of better understanding rapid changes in seed dispersal and their consequences for the reaction of plant populations to global change., Global ecological change is causing plant populations to either adapt or move in response to new environmental conditions. In many species it is thought that seed dispersal may not allow populations to move at a rate that tracks the shifting climate. However, phenotypic plasticity and rapid genetic changes in traits associated with seed dispersal may modify the response capabilities of plants and allow them to responds more quickly. We explore the evidence for rapid modification of seed dispersal traits and propose a path forward to better understand the ways in which seed dispersal may buffer the effects of global ecological change.

We study the evolutionary stability of dispersal strategies, including but not limited to those that can produce ideal free population distributions (that is, distributions where all individuals have equal fitness and there is no net movement of individuals at equilibrium). The environment is assumed to be variable in space but constant in time. We assume that there is a separation of times scales, so that dispersal occurs on a fast timescale, evolution occurs on a slow timescale, and population dynamics and interactions occur on an intermediate timescale. Starting with advection–diffusion models for dispersal without population dynamics, we use the large time limits of profiles for population distributions together with the distribution of resources in the environment to calculate growth and interaction coefficients in logistic and Lotka–Volterra ordinary differential equations describing population dynamics. We then use a pairwise invasibility analysis approach motivated by adaptive dynamics to study the evolutionary and/or convergence stability of strategies determined by various assumptions about the advection and diffusion terms in the original advection–diffusion dispersal models. Among other results we find that those strategies which can produce an ideal free distribution are evolutionarily stable.

Traditionally, ecologists interested in habitat edges have focused on edge-related gradients in patterns of species richness or abiotic variables. Here, however, we take a different perspective, attempting to synthesize recent empirical results concerning the effects of habitat edges on population dynamics with contemporary theoretical developments to outline the ways in which species interactions, and the dynamics of the communities in which they are embedded, can be changed by habitat edges. We find a striking convergence between empirical notions of a patch's core area and analytical results from partial differential equation models. A review of both empirical and theoretical studies suggests four general classes of mechanisms through which habitat edges can have similar impacts on dissimilar types of species interactions. Specifically, we focus on edges' roles as dispersal barriers or filters, edges' influences on mortality, edges' involvement in spatial subsidies (in which dispersers' intrapatch impacts are maintained by their activities in other habitats), and edges' roles as generators of novel interactions. For each class of edge-mediated effects, we provide examples of how one can use spatial modeling to address the relevant questions on these topics, which together form a key link between community dynamics and landscape structure.

A cross-diffusion model of an intraguild predation community in a two-dimensional bounded domain where the intraguild prey employs a fitness based avoidance strategy is examined. The avoidance strategy employed is to increase motility in response to negative local fitness. Global existence of trajectories and the existence of a compact global attractor is proved. It is shown that if the intraguild prey has positive fitness at any point in the habitat when trying to invade, then it will be uniformly persistent in the system if its avoidance tendency is sufficiently strong. This type of movement strategy can lead to coexistence states where the intraguild prey is marginalized to areas with low resource productivity while the intraguild predator maintains high densities in regions with abundant resources, a pattern observed in many real world intraguild predation systems. Additionally, the effects of fitness based avoidance on eigenvalues in more general systems are discussed.

How organisms gather and utilize information about their landscapes is central to understanding land-use patterns and population distributions. When such information originates beyond an individual's immediate vicinity, movement decisions require integrating information out to some perceptual range. Such nonlocal information, whether obtained visually, acoustically, or via chemosensation, provides a field of stimuli that guides movement. Classically, however, models have assumed movement based on purely local information (e.g., chemotaxis, step-selection functions). Here we explore how foragers can exploit nonlocal information to improve their success in dynamic landscapes. Using a continuous time/continuous space model in which we vary both random (diffusive) movement and resource-following (advective) movement, we characterize the optimal perceptual ranges for foragers in dynamic landscapes. Nonlocal information can be highly beneficial, increasing the spatiotemporal concentration of foragers on their resources up to twofold compared with movement based on purely local information. However, nonlocal information is most useful when foragers possess both high advective movement (allowing them to react to transient resources) and low diffusive movement (preventing them from drifting away from resource peaks). Nonlocal information is particularly beneficial in landscapes with sharp (rather than gradual) patch edges and in landscapes with highly transient resources.

This work extends previous work (Cantrell et al., 2008 [9] ) on fitness-dependent dispersal for a single species to a two-species competition model. Both species have the same population dynamics, but one species adopts a combination of random and fitness-dependent dispersal and the other adopts random dispersal. Global existence of smooth solutions to the time-dependent quasilinear parabolic system is studied. When a single species has a strong tendency to move up its fitness gradient, it results in a stable equilibrium that can approximate the spatial distribution predicted by the ideal free distribution (Cantrell et al., 2008 [9] ). For the two-species competition model, if one species has strong tendency to move up its fitness gradient, such approximately ideal free dispersal is advantageous relative to random dispersal. Bifurcation analysis shows that two competing species can coexist when one species has only an intermediate tendency to move up its fitness gradient and the other species has a smaller random dispersal rate.

Understanding the evolution of dispersal is an important issue in evolutionary ecology. For continuous time models in which individuals disperse throughout their lifetime, it has been shown that a balanced dispersal strategy, which results in an ideal free distribution, is evolutionary stable in spatially varying but temporally constant environments. Many species, however, primarily disperse prior to reproduction (natal dispersal) and less commonly between reproductive events (breeding dispersal). As demographic and dispersal terms combine in a multiplicative way for models of natal dispersal, rather than the additive way for the previously studied models, we develop new mathematical methods to study the evolution of natal dispersal for continuous-time and discrete-time models. A fundamental ecological dichotomy is identified for the non-trivial equilibrium of these models: (i) the per-capita growth rates for individuals in all patches is equal to zero, or (ii) individuals in some patches experience negative per-capita growth rates, while individuals in other patches experience positive per-capita growth rates. The first possibility corresponds to an ideal-free distribution, while the second possibility corresponds to a "source-sink" spatial structure. We prove that populations with a dispersal strategy leading to an ideal-free distribution displace populations with dispersal strategy leading to a source-sink spatial structure. When there are patches which can not sustain a population, ideal-free strategies can be achieved by sedentary populations, and we show that these populations can displace populations with any irreducible dispersal strategy. Collectively, these results support that evolution selects for natal or breeding dispersal strategies which lead to ideal-free distributions in spatially heterogenous, but temporally homogenous, environments., Comment: Revision correcting several minor errors

We study pattern formation in a quasi-linear system of two elliptic equations that was developed by Short et al. [Math. Models Methods Appl. Sci., 18 (2008), pp. 1249–1267] as a model for residential burglary. That model is based on the observation that the rate of burglaries of houses that have been burglarized recently and their close neighbors is typically higher than the average rate in the larger community, which creates hotspots for burglary. The patterns generated by the model describe the location of those hotspots. We prove that the system supports global bifurcation of spatially varying solutions from the spatially constant equilibrium, leading to the formation of spatial patterns. The analysis is based on recent results on global bifurcation in quasi-linear elliptic systems derived by Shi and Wang [J. Differential Equations, 7 (2009), pp. 2788–2812]. We show in some cases that near the bifurcation point the bifurcating spatial patterns are stable.

In this article we compare and contrast the predictions of some spatially explicit and implicit models in the context of a thought problem at the interface of spatial and landscape ecology. The situation we envision is a one-dimensional spatial universe of infinite extent in which there are two disjoint focal patches of a habitat type that is favorable to some specified species. We assume that neither patch is large enough by itself to sustain the species in question indefinitely, but that a single patch of size equal to the combined sizes of the two focal patches provides enough contiguous favorable habitat to sustain the given species indefinitely. When the two patches are separated by a patch of unfavorable matrix habitat, the natural expectation is that the species should persist indefinitely if the two patches are close enough to each other but should go extinct over time when the patches are far enough apart. Our focus here is to examine how different mathematical regimes may be employed to model this situation, with an eye toward exploring the trade-off between the mathematical tractability of the model on one hand and the suitability of its predictions on the other. In particular, we are interested in seeing how precisely the predictions of mathematically rich spatially explicit regimes (reaction-diffusion models, integro-difference models) can be matched by those of ostensibly mathematically simpler spatially implicit patch approximations (discrete-diffusion models, average dispersal success matrix models).

Long-distance animal migrations are complex, population-level phenomena that emerge in seasonal landscapes as a result of the interplay between environmental influences (e.g., resources, predators) and social interactions among conspecifics. When landscapes change with respect to phenology or connectivity, the dynamics of migratory species can abruptly shift, in many cases leading to a cessation of migration and dramatic decreases in population size. We develop a difference equation modeling framework to explore how the social transfer of knowledge from informed “leader” individuals enhances the performance of seasonally migratory versus resident populations. The model permits a wide range of population-level behaviors including alternative stable states, partial migration equilibria, and complex dynamics, but we focus our efforts on investigations of migration collapse mediated by a lack of informed leaders that can arise from changes in landscape structure, survivorship, reproduction, and/or social learning. Migration collapse is a hysteretic phenomenon in this model and results either in extinction of the population or purely resident behavior. The hysteretic nature of migration failure, which hinges on cultural transmission of knowledge, highlights a potentially critical role for behavior and social learning in aspects of spatial ecology and conservation biology.

A general question in the study of the evolution of dispersal is what kind of dispersal strategies can convey competitive advantages and thus will evolve. We consider a two species competition model in which the species are assumed to have the same population dynamics but different dispersal strategies. Both species disperse by random diffusion and advection along certain gradients, with the same random dispersal rates but different advection coefficients. We found a conditional dispersal strategy which results in the ideal free distribution of species, and show that it is a local evolutionarily stable strategy. We further show that this strategy is also a global convergent stable strategy under suitable assumptions, and our results illustrate how the evolution of conditional dispersal can lead to an ideal free distribution. The underlying biological reason is that the species with this particular dispersal strategy can perfectly match the environmental resource, which leads to its fitness being equilibrated across the habitats.

With the recent resurgence of vector-borne diseases due to urbanization and development there is an urgent need to understand the dynamics of vector-borne diseases in rapidly changing urban environments. For example, many empirical studies have produced the disturbing finding that diseases continue to persist in modern city centers with zero or low rates of transmission. We develop spatial models of vector-borne disease dynamics on a network of patches to examine how the movement of humans in heterogeneous environments affects transmission. We show that the movement of humans between patches is sufficient to maintain disease persistence in patches with zero transmission. We construct two classes of models using different approaches: (i) Lagrangian models that mimic human commuting behavior and (ii) Eulerian models that mimic human migration. We determine the basic reproduction number R(0) for both modeling approaches. We show that for both approaches that if the disease-free equilibrium is stable (R(0)1) then there exists a unique positive (endemic) equilibrium that is globally stable among positive solutions. Finally, we prove in general that Lagrangian and Eulerian modeling approaches are not equivalent. The modeling approaches presented provide a framework to explore spatial vector-borne disease dynamics and control in heterogeneous environments. As an example, we consider two patches in which the disease dies out in both patches when there is no movement between them. Numerical simulations demonstrate that the disease becomes endemic in both patches when humans move between the two patches. National Institutes of Health/[P20-RR020770]/NIH/Estados Unidos National Science Foundation/[DMS-0514839]/NSF/Estados Unidos National Science Foundation/[DMS-0816068]/NSF/Estados Unidos National Science Foundation/[DMS-0715772]/NSF/Estados Unidos UCR::Vicerrectoría de Investigación::Unidades de Investigación::Ciencias de la Salud::Centro de Investigación en Enfermedades Tropicales (CIET)

A food chain consisting of species at three trophic levels is modeled using Beddington-DeAngelis functional responses as the links between trophic levels. The dispersal of the species is modeled by diffusion, so the resulting model is a three component reaction-diffusion system. The behavior of the system is described in terms of predictions of extinction or persistence of the species. Persistence is characterized via permanence, i.e., uniform persistence plus dissi-pativity. The way that the predictions of extinction or persistence depend on domain size is studied by examining how they vary as the size (but not the shape) of the underlying spatial domain is changed.

We examine the evolutionary stability of strategies for dispersal in heterogeneous patchy environments or for switching between discrete states (e.g. defended and undefended) in the context of models for population dynamics or species interactions in either continuous or discrete time. There have been a number of theoretical studies that support the view that in spatially heterogeneous but temporally constant environments there will be selection against unconditional, i.e. random, dispersal, but there may be selection for certain types of dispersal that are conditional in the sense that dispersal rates depend on environmental factors. A particular type of dispersal strategy that has been shown to be evolutionarily stable in some settings is balanced dispersal, in which the equilibrium densities of organisms on each patch are the same whether there is dispersal or not. Balanced dispersal leads to a population distribution that is ideal free in the sense that at equilibrium all individuals have the same fitness and there is no net movement of individuals between patches or states. We find that under rather general assumptions about the underlying population dynamics or species interactions, only such ideal free strategies can be evolutionarily stable. Under somewhat more restrictive assumptions (but still in considerable generality), we show that ideal free strategies are indeed evolutionarily stable. Our main mathematical approach is invasibility analysis using methods from the theory of ordinary differential equations and nonnegative matrices. Our analysis unifies and extends previous results on the evolutionary stability of dispersal or state-switching strategies.

We study a Lotka–Volterra reaction–diffusion–advection model for two competing species in a heterogeneous environment. The species are assumed to be identical except for their dispersal strategies: one disperses by random diffusion only, the other by both random diffusion and advection along an environmental gradient. When the two competitors have the same diffusion rates and the strength of the advection is relatively weak in comparison to that of the random dispersal, we show that the competitor that moves towards more favourable environments has the competitive advantage, provided that the underlying spatial domain is convex, and the competitive advantage can be reversed for certain non-convex habitats. When the advection is strong relative to the dispersal, we show that both species can invade when they are rare, and the two competitors can coexist stably. The biological explanation is that, for sufficiently strong advection, the ‘smarter' competitor will move towards more favourable environments and is concentrated at the place with maximum resources. This leaves enough room for the other species to survive, since it can live upon regions with finer quality resources.

Rift Valley fever (RVF) is an important mosquito-borne viral zoonosis in Africa and the Middle East that causes human deaths and significant economic losses due to huge incidences of death and abortion among infected livestock. Outbreaks of RVF are sporadic and associated with both seasonal and socioeconomic effects. Here we propose an almost periodic three-patch model to investigate the transmission dynamics of RVF virus (RVFV) among ruminants with spatial movements. Our findings indicate that, in Northeastern Africa, human activities, including those associated with the Eid al Adha feast, along with a combination of climatic factors such as rainfall level and hydrological variations, contribute to the transmission and dispersal of the disease pathogen. Moreover, sporadic outbreaks may occur when the two events occur together: 1) abundant livestock are recruited into areas at risk from RVF due to the demand for the religious festival and 2) abundant numbers of mosquitoes emerge. These two factors have been shown to have impacts on the severity of RVF outbreaks. Our numerical results present the transmission dynamics of the disease pathogen over both short and long periods of time, particularly during the festival time. Further, we investigate the impact on patterns of disease outbreaks in each patch brought by festival- and seasonal-driven factors, such as the number of livestock imported daily, the animal transportation speed from patch to patch, and the death rate induced by ceremonial sacrifices. In addition, our simulations show that when the time for festival preparation starts earlier than usual, the risk of massive disease outbreaks rises, particularly in patch 3 (the place where the religious ceremony will be held)., Author Summary Rift Valley fever is a common vector-borne zoonotic disease that causes huge economic losses in Africa and the Middle East. The transmission and dispersal of the disease pathogen are affected by many factors, such as climatic, hydrologic and geographic influences, along with impacts from human activities and different forms of virus transmission via different vectors. In this work, we focus on identifying the potential risks that lead to disease outbreaks in Egypt along the Nile, from the South to the Nile Delta, by mathematically and numerically analyzing a patch model with temporal periodicity. We discover that human activities during the Eid al Adha feast, as well as climatic and hydrological variations, contribute to the transmission and dispersal of the disease. Interestingly, periodic co-occurrence of the religious festival and the onset of peaks in mosquito abundance, each with a different periodicity, is predicted to lead to periodic large scale disease outbreaks.

We study a diffusive logistic equation with nonlinear boundary conditions. The equation arises as a model for a population that grows logistically inside a patch and crosses the patch boundary at a rate that depends on the population density. Specifically, the rate at which the population crosses the boundary is assumed to decrease as the density of the population increases. The model is motivated by empirical work on the Glanville fritillary butterfly. We derive local and global bifurcation results which show that the model can have multiple equilibria and in some parameter ranges can support Allee effects. The analysis leads to eigenvalue problems with nonstandard boundary conditions.

Empirical data for several ecological systems suggest that how resource availability scales with patch geometry may influence the outcome of species interactions. To study this process, we assume a pseudoequilibrium to reduce the dimensionality of a two-consumer-two-resource model in which different resources are available in the interior of a patch versus at the edge. We analyze the resulting two species competition model to understand how the outcome of competition between consumers changes as the size of the patch changes, paying particular attention to the differential scaling of interior and edge-linked allochthonous resources as a function of patch size. We characterize conditions on patch size and parameters under which competitive exclusion, coexistence, and a reversal in competitive dominance occur. We find that the degree of exclusivity in the use of edge versus interior habitats influences the potential for transitions in competitive outcomes, but that differences in resource quality between interior and edge habitats can, depending on the scenario, have either qualitative or quantitative influences on the transitions. The work highlights the importance of patch size to understanding species interactions and demonstrates that competitive dominance can be a scale- dependent trait.