Ideal free dispersal in integrodifference models.

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.

Evolutionarily stable movement strategies in reaction-diffusion models with edge behavior.

Many types of organisms disperse through heterogeneous environments as part of their life histories. For various models of dispersal, including reaction-advection-diffusion models in continuously varying environments, it has been shown by pairwise invasibility analysis that dispersal strategies which generate an ideal free distribution are evolutionarily steady strategies (ESS, also known as evolutionarily stable strategies) and are neighborhood invader strategies (NIS). That is, populations using such strategies can both invade and resist invasion by populations using strategies that do not produce an ideal free distribution. (The ideal free distribution arises from the assumption that organisms inhabiting heterogeneous environments should move to maximize their fitness, which allows a mathematical characterization in terms of fitness equalization.) Classical reaction diffusion models assume that landscapes vary continuously. Landscape ecologists consider landscapes as mosaics of patches where individuals can make movement decisions at sharp interfaces between patches of different quality. We use a recent formulation of reaction-diffusion systems in patchy landscapes to study dispersal strategies by using methods inspired by evolutionary game theory and adaptive dynamics. Specifically, we use a version of pairwise invasibility analysis to show that in patchy environments, the behavioral strategy for movement at boundaries between different patch types that generates an ideal free distribution is both globally evolutionarily steady (ESS) and is a global neighborhood invader strategy (NIS).

Evolution of dispersal in spatial population models with multiple timescales.

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.

Habitat fragmentation promotes malaria persistence.

Based on a Ross-Macdonald type model with a number of identical patches, we study the role of the movement of humans and/or mosquitoes on the persistence of malaria and many other vector-borne diseases. By using a theorem on line-sum symmetric matrices, we establish an eigenvalue inequality on the product of a class of nonnegative matrices and then apply it to prove that the basic reproduction number of the multipatch model is always greater than or equal to that of the single patch model. Biologically, this means that habitat fragmentation or patchiness promotes disease outbreaks and intensifies disease persistence. The risk of infection is minimized when the distribution of mosquitoes is proportional to that of humans. Numerical examples for the two-patch submodel are given to investigate how the multipatch reproduction number varies with human and/or mosquito movement. The reproduction number can surpass any given value whenever an appropriate travel pattern is chosen. Fast human and/or mosquito movement decreases the infection risk, but may increase the total number of infected humans.

Modeling the importation and local transmission of vector-borne diseases in Florida: The case of Zika outbreak in 2016.

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.

Two-patch model for the spread of West Nile virus.

A two-patch model for the spread of West Nile virus between two discrete geographic regions is established to incorporate a mobility process which describes how contact transmission occurs between individuals from and between two regions. In the mobility process, we assume that the host birds can migrate between regions, but not the mosquitoes. The basic reproduction number [Formula: see text] is computed by the next generation matrix method. We prove that if [Formula: see text], then the disease-free equilibrium is globally asymptotically stable. If [Formula: see text], the endemic equilibrium is globally asymptotically stable for any nonnegative nontrivial initial data. Using the perturbation theory, we obtain the concrete expression of the endemic equilibrium of the model with a mild restriction of the birds movement rate between patches. Finally, numerical simulations demonstrate that the disease becomes endemic in both patches when birds move back and forth between the two regions. Some numerical simulations for [Formula: see text] in terms of the birds movement rate are performed which show that the impacts could be very complicated.

Perceptual Ranges, Information Gathering, and Foraging Success in Dynamic Landscapes.

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.

How Resource Phenology Affects Consumer Population Dynamics.

Climate change drives uneven phenology shifts across taxa, and this can result in changes to the phenological match between interacting species. Shifts in the relative phenology of partner species are well documented, but few studies have addressed the effects of such changes on population dynamics. To explore this, we develop a phenologically explicit model describing consumer-resource interactions. Focusing on scenarios for univoltine insects, we show how changes in resource phenology can be reinterpreted as transformations in the year-to-year recursion relationships defining consumer population dynamics. This perspective provides a straightforward path for interpreting the long-term population consequences of phenology change. Specifically, by relating the outcome of phenological shifts to species traits governing recursion relationships (e.g., consumer fecundity or competitive scenario), we demonstrate how changes in relative phenology can force systems into different dynamical regimes, with major implications for resource management, conservation, and other areas of applied dynamics.

A modeling approach to investigate epizootic outbreaks and enzootic maintenance of Rift Valley fever virus.

We propose a mathematical model to investigate the transmission dynamics of Rift Valley fever (RVF) virus among ruminants. Our findings indicate that in endemic areas RVF virus maintains at a very low level among ruminants after outbreaks and subsequent outbreaks may occur when new susceptible ruminants are recruited into endemic areas or abundant numbers of mosquitoes emerge when herd immunity decreases. Many factors have been shown to have impacts on the severity of RVF outbreaks; a higher probability of death due to RVF among ruminants, a higher mosquito:ruminant ratio, or a shorter lifespan of animals can amplify the magnitude of the outbreaks; vaccination helps to reduce the magnitude of RVF outbreaks and the loss of animals efficiently, and the maximum vaccination effort (a high vaccination rate and a larger number of vaccinated animals) is recommended before the commencement of an outbreak but can be reduced later during the enzootic.

Well-posedness and qualitative properties of a dynamical model for the ideal free distribution.

Understanding the spatial distribution of populations in heterogeneous environments is an important problem in ecology. In the case of a population of organisms that can sense the quality of their environment and move to increase their fitness, one theoretical description of the expected distribution of the population is the ideal free distribution, where individuals locate themselves to optimize fitness. A model for a dynamical process that allows a population to achieve an ideal free distribution was proposed by the Cosner (Theor Popul Biol 67:101-108, 2005). The model is based on a reaction-diffusion-advection equation with nonlinear diffusion which is similar to a porous medium equation with additional advection and population growth terms. We establish that the model is well-posed, show that solutions stabilize, determine the stationary states, discuss their stability, and describe the biological interpretation of the results.

Modeling the spatial spread of Rift Valley fever in Egypt.

Rift Valley fever (RVF) is a severe viral zoonosis in Africa and the Middle East that harms both human health and livestock production. It is believed that RVF in Egypt has been repeatedly introduced by the importation of infected animals from Sudan. In this paper, we propose a three-patch model for the process by which animals enter Egypt from Sudan, are moved up the Nile, and then consumed at population centers. The basic reproduction number for each patch is introduced and then the threshold dynamics of the model are established. We simulate an interesting scenario showing a possible explanation of the observed phenomenon of the geographic spread of RVF in Egypt.

On the Allee effect and directed movement on the whole space.

It is well known that relocation strategies in ecology can make the difference between extinction and persistence. We consider a reaction-advection-diffusion framework to analyze movement strategies in the context of species which are subject to a strong Allee effect. The movement strategies we consider are a combination of random Brownian motion and directed movement through the use of an environmental signal. We prove that a population can overcome the strong Allee effect when the signals are super-harmonic. In other words, an initially small population can survive in the long term if they aggregate sufficiently fast. A sharp result is provided for a specific signal that can be related to the Fokker-Planck equation for the Orstein-Uhlenbeck process. We also explore the case of pure diffusion and pure aggregation and discuss their benefits and drawbacks, making the case for a suitable combination of the two as a better strategy.

Linking mathematical models and trap data to infer the proliferation, abundance, and control of Aedes aegypti.

Aedes aegypti is one of the most dominant mosquito species in the urban areas of Miami-Dade County, Florida, and is responsible for the local arbovirus transmissions. Since August 2016, mosquito traps have been placed throughout the county to improve surveillance and guide mosquito control and arbovirus outbreak response. In this paper, we develop a deterministic mosquito population model, estimate model parameters by using local entomological and temperature data, and use the model to calibrate the mosquito trap data from 2017 to 2019. We further use the model to compare the Ae. aegypti population and evaluate the impact of rainfall intensity in different urban built environments. Our results show that rainfall affects the breeding sites and the abundance of Ae. aegypti more significantly in tourist areas than in residential places. In addition, we apply the model to quantitatively assess the effectiveness of vector control strategies in Miami-Dade County.

Evolutionary stability of ideal free dispersal strategies in patchy environments.

A central question in the study of the evolution of dispersal is what kind of dispersal strategies are evolutionarily stable. Hastings (Theor Pop Biol 24:244-251, 1983) showed that among unconditional dispersal strategies in a spatially heterogeneous but temporally constant environment, the dispersal strategy with no movement is convergent stable. McPeek and Holt's (Am Nat 140:1010-1027, 1992) work suggested that among conditional dispersal strategies in a spatially heterogeneous but temporally constant environment, an ideal free dispersal strategy, which results in the ideal free distribution for a single species at equilibrium, is evolutionarily stable. We use continuous-time and discrete-space models to determine when the dispersal strategy with no movement is evolutionarily stable and when an ideal free dispersal strategy is evolutionarily stable, both in a spatially heterogeneous but temporally constant environment.

Emergence and dynamics of influenza super-strains.

BACKGROUND: Influenza super-strains can emerge through recombination of strains from birds, pigs, and humans. However, once a new recombinant strain emerges, it is not clear whether the strain is capable of sustaining an outbreak. In certain cases, such strains have caused major influenza pandemics. METHODS: Here we develop a multi-host (i.e., birds, pigs, and humans) and multi-strain model of influenza to analyze the outcome of emergent strains. In the model, pigs act as "mixing vessels" for avian and human strains and can produce super-strains from genetic recombination. RESULTS: We find that epidemiological outcomes are predicted by three factors: (i) contact between pigs and humans, (ii) transmissibility of the super-strain in humans, and (iii) transmissibility from pigs to humans. Specifically, outbreaks will reoccur when the super-strain infections are less frequent between humans (e.g., R0=1.4) but frequent from pigs to humans, and a large-scale outbreak followed by successively damping outbreaks will occur when human transmissibility is high (e.g., R0=2.3). The average time between the initial outbreak and the first resurgence varies from 41 to 82 years. We determine the largest outbreak will occur when 2.3

Reproductive asynchrony in spatial population models: how mating behavior can modulate allee effects arising from isolation in both space and time.

Mate finding, which is essential to both population growth and gene exchange, involves both spatial and temporal components. From a population dynamics perspective, spatial mate-finding problems are well studied, and decreased mate-finding efficiency at low population densities is a well-recognized mechanism for the Allee effect. Temporal aspects of mate finding have been rarely considered, but reproductive asynchrony may engender an Allee effect in which some females go mateless by virtue of temporal isolation. Here we develop and explore a model that unifies previously disparate theoretical considerations of spatial and temporal aspects of mate finding. Specifically, we develop a two-sex reaction-diffusion system to examine the interplay between reproductive asynchrony and the dispersal of individuals out of a patch. We also consider additional behavioral complications, including several alternative functional forms for mating efficiency and advective movements in which males actively seek out females. By calculating the fraction of females expected to go mateless as a joint function of reproductive asynchrony and patch size, we find that the population-level reproductive rates necessary to offset female matelessness may be quite high. These results suggest that Allee effects engendered by reproductive asynchrony will be greatly exacerbated in spatially isolated populations.

Interspecific variation in critical patch size and gap-crossing ability as determinants of geographic range size distributions.

How biological processes such as reproduction and dispersal relate to the size of species' geographic ranges constitutes a major challenge in spatial ecology and biogeography. Here we develop a spatially explicit theoretical framework that links fundamental population-level ecological traits (e.g., rates of dispersal and population growth or decay) with landscape heterogeneity to derive estimates of species' geographic range sizes and, further, distributions of geographic range sizes across species. Although local (patch-scale) population dynamics in this model are completely deterministic, we consider a fragmented landscape of patches and gaps in which the spatial heterogeneity is itself stochastic. This stochastic spatial structure, which juxtaposes landscape-level patch and gap characteristics against population-level critical patch sizes and maximum gap-crossing abilities, determines how far a novel species can spread from its evolutionary origin. Given reasonable assumptions about landscape structure and about the distribution of critical patch sizes and critical gap lengths among species, we obtain distributions of geographic range sizes that are qualitatively similar to those routinely found in nature (e.g., many species with small geographic ranges). Collectively, our results suggest that both interspecific differences in population-level traits and the landscapes through which species spread help determine patterns of occupancy and geographic extent.

Rapid changes in seed dispersal traits may modify plant responses to global change.

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.

Evolutionary stability of ideal free dispersal under spatial heterogeneity and time periodicity.

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.

Dynamics of populations with individual variation in dispersal on bounded domains.

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.