Competition in a spatially heterogeneous environment: modelling the risk of spread of a genetically engineered population

Polymorphic population expansion velocity in a heterogeneous environment

How does the spatial heterogeneity of landscapes interact with the adaptive evolution of populations to influence their spreading speed? This question arises in agricultural contexts where a pathogen population spreads in a landscape composed of several types of crops, as well as in epidemiological settings where a virus spreads among individuals with distinct immune profiles. To address it, we introduce an analytical method based on reaction-diffusion models. We focus on spatially periodic environments with two distinct patches, where the dispersing population consists of two specialized morphs, each potentially mutating to the other. We present new formulas for the speed together with criteria for persistence, accounting for both rapidly and slowly varying environments, as well as small and large mutation rates. Altogether, our analytical and numerical results yield a comprehensive understanding of persistence and spreading dynamics. In particular, compared to a situation without mutations or to a single morph spreading in a heterogeneous landscape, the introduction of mutations to a second morph with reverse specialization, while consistently impeding persistence, can significantly increase speed, even if the mutation rate between the two morphs is very small. Additionally, we find that the amplitude of the spatial fragmentation effect is significantly increased in this case. This has implications for agroecology, emphasizing the higher importance of landscape structure in influencing adaptation-driven population dynamics.

Persistence and spread of stage-structured populations in heterogeneous landscapes

Conditions for population persistence in heterogeneous landscapes and formulas for population spread rates are important tools for conservation ecology and invasion biology. To date, these tools have been developed for unstructured populations, yet many, if not all, species show two or more distinct phases in their life cycle. We formulate and analyze a stage-structured model for a population in a heterogeneous habitat. We divide the population into pre-reproductive and reproductive stages. We consider an environment consisting of two types of patches, one where population growth is positive, one where it is negative. Individuals move randomly within patches but can show preference towards one patch type at the interface between patches. We use linear stability analysis to determine persistence conditions, and we derive a dispersion relation to find spatial spread rates. We illustrate our results by comparing the structured population model with an appropriately scaled unstructured model. We find that a long pre-reproductive state typically increases habitat requirements for persistence and decreases spatial spread rates, but we also identify scenarios in which a population with intermediate maturation rate spreads fastest.

Propagation of fronts in the Fisher-Kolmogorov equation with spatially varying diffusion

The propagation of fronts in the Fisher-Kolmogorov equation with spatially varying diffusion coefficients is studied. Using coordinate changes, WKB approximations, and multiple scales analysis, we provide an analytic framework that describes propagation of the front up to the minimum of the diffusion coefficient. We also present results showing the behavior of the front after it passes the minimum. In each case, we show that standard traveling coordinate frames do not properly describe front propagation. Last, we provide numerical simulations to support our analysis and to show, that around the minimum, the motion of the front is arrested on asymptotically significant time scales.

How flow speed alters competitive outcome in advective environments

Many species live in advective environments, such as rivers or streams. The community composition in such environments is shaped by the interplay between biotic interactions and hydrologic constraints. Lutscher et al. (Theor. Popul. Biol. 71:267-277, 2007) demonstrated by simulation that advective flow can shift competitive outcome from one species dominating to coexistence or even to the other species dominating. Here, we present a detailed analysis of the Lotka-Volterra advection-diffusion model underlying their simulations. We use variational techniques as well as a spatially implicit approximation to determine all possible advection-induced shifts in competitive outcome. We show that changes in advection follow relatively few and predictable paths. We illustrate our results in various bifurcation diagrams.

The effect of the spatial configuration of habitat fragmentation on invasive spread

To address how the spatial configuration of habitat fragmentation influences the persistence and the rate of spread of an invasive species, we consider three simple periodically fragmented environments, a lattice-like corridor environment, an island-like environment and a striped environment. By numerically analyzing Fisher's equation with a spatially varying diffusion coefficient and the intrinsic growth rate, we find the following. (1) When the scale of fragmentation is sufficiently large, the minimum favorable area needed for successful invasion reduces in the following order: lattice-like corridor, striped and island-like environments. (2) When the scale of fragmentation and the fraction of favorable area are sufficiently large, the spreading speeds along contiguous favorable habitats in the lattice-like corridor and striped environments are faster than the speeds across isolated favorable habitats in the island-like environment and the striped environment. (3) When the periodicity of fragmentation is relaxed by stochastically shifting the boundaries between favorable and unfavorable habitats, the average speed increases with increases in the irregularity of fragmentation.

Dispersal in heterogeneous habitats: thresholds, spatial scales, and approximate rates of spread

What is the effect of landscape heterogeneity on the spread rate of populations? Several spatially explicit simulation models address this question for particular cases and find qualitative insights (e.g., extinction thresholds) but no quantitative relationships. We use a time-discrete analytic model and find general quantitative relationships for the invasion threshold, i.e., the minimal percentage of suitable habitat required for population spread. We investigate how, on the relevant spatial scales, this threshold depends on the relationship between dispersal ability and fragmentation level. The invasion threshold increases with fragmentation level when there is no Allee effect, but it decreases with fragmentation in the presence of an Allee effect. We obtain simple formulas for the approximate spread rate of a population in heterogeneous landscapes from averaging techniques. Comparison with spatially explicit simulations shows an excellent agreement between approximate and true values. We apply our results to the spread of trees and give some implications for the control of invasive species.

Evolutionary epidemiology of drug-resistance in space

How can we optimize the use of drugs against parasites to limit the evolution of drug resistance? This question has been addressed by many theoretical studies focusing either on the mixing of various treatments, or their temporal alternation. Here we consider a different treatment strategy where the use of the drug may vary in space to prevent the rise of drug-resistance. We analyze epidemiological models where drug-resistant and drug-sensitive parasites compete in a one-dimensional spatially heterogeneous environment. Two different parasite life-cycles are considered: (i) direct transmission between hosts, and (ii) vector-borne transmission. In both cases we find a critical size of the treated area, under which the drug-resistant strain cannot persist. This critical size depends on the basic reproductive ratios of each strain in each environment, on the ranges of dispersal, and on the duration of an infection with drug-resistant parasites. We discuss optimal treatment strategies that limit disease prevalence and the evolution of drug-resistance.

The spread of drug-resistant parasites erodes the efficacy of therapeutic treatments against many infectious diseases and is a major threat of the 21st century. The evolution of drug-resistance depends, among other things, on how the treatments are administered at the population level. “Resistance management” consists of finding optimal treatment strategies that both reduce the consequence of an infection at the individual host level, and limit the spread of drug-resistance in the pathogen population. Several studies have focused on the effect of mixing different treatments, or of alternating them in time. Here, we analyze another strategy, where the use of the drug varies spatially: there are places where no one receives any treatment. We find that such a spatial heterogeneity can totally prevent the rise of drug-resistance, provided that the size of treated patches is below a critical threshold. The range of parasite dispersal, the relative costs and benefits of being drug-resistant compared to being drug-sensitive, and the duration of an infection with drug-resistant parasites are the main factors determining the value of this threshold. Our analysis thus provides some general guidance regarding the optimal spatial use of drugs to prevent or limit the evolution of drug-resistance.

Density-dependent dispersal in integrodifference equations

Many species exhibit dispersal processes with positive density- dependence. We model this behavior using an integrodifference equation where the individual dispersal probability is a monotone increasing function of local density. We investigate how this dispersal probability affects the spreading speed of a single population and its ability to persist in fragmented habitats. We demonstrate that density-dependent dispersal probability can act as a mechanism for coexistence of otherwise non-coexisting competitors. We show that in time-varying habitats, an intermediate dispersal probability will evolve. Analytically, we find that the spreading speed for the integrodifference equation with density-dependent dispersal probability is not linearly determined. Furthermore, the next-generation operator is not compact and, in general, neither order-preserving nor monotonicity-preserving. We give two explicit examples of non-monotone, discontinuous traveling-wave profiles.

Effects of Heterogeneity on Spread and Persistence in Rivers

The question how aquatic populations persist in rivers when individuals are constantly lost due to downstream drift has been termed the "drift paradox." Recent modeling approaches have revealed diffusion-mediated persistence as a solution. We study logistically growing populations with and without a benthic stage and consider spatially varying growth rates. We use idealized hydrodynamic equations to link river cross-sectional area to flow speed and assume heterogeneity in the form of alternating patches, i.e., piecewise constant conditions. We derive implicit formulae for the persistence boundary and for the dispersion relation of the wave speed. We explicitly discuss the influence of flow speed, cross-sectional area and benthic stage on both persistence and upstream invasion speed.

Competition and dispersal in predator-prey waves

Dispersing predators and prey can exhibit complex spatio-temporal wave-like patterns if the interactions between them cause oscillatory dynamics. We study the effect of these predator-prey density waves on the competition between prey populations and between predator populations with different dispersal strategies. We first describe 1- and 2-dimensional simulations of both discrete and continuous predator-prey models. The results suggest that any population that diffuses faster, disperses farther, or is more likely to disperse will exclude slower diffusing, shorter dispersing, or less likely dispersing populations, everything else being equal. It also appears that it does not matter whether time, space, or state are discrete or continuous, nor what the exact interactions between the predators and prey are. So long as waves exist the competition between populations occurs in a similar fashion. We derive a theory that qualitatively explains the observed behaviour and calculate approximate analytical solutions that describe, to a reasonable extent, these behaviours. Predictions about the cost of dispersal are tested. If strong enough, cost can reverse the populations' relative competitive strengths or lead to coexistence because of the effect of spiral wave cores. The theory is also able to explain previous results of simulations of coexistence in host-parasitoid models (Comins, H. N., and Massell, M. P., 1996, J. Theor. Biol. 183, 19-28).

A sex-structured delayed recruitment model

A delay-difference model for a sex-structured population with delayed recruitment is presented. Constant-effort harvesting is introduced for examining the model's sensitivity to harvesting. Linear analyses about the steady states are performed under various parameter choices. The effects of the delay period, the survival parameter, and the harvesting effort on population stability are examined. It is shown that differences in the male and female delays to recruitment can give rise to very different stability diagrams. Our analyses indicate, for example, that a population with equal male and female delays to recruitment is the most robust to recruitment failures. Possible forms that the male and female recruitment functions could take are suggested and evaluated. Finally, the very encouraging result is obtained that maximum sustainable yield is attained at a stable steady-state population level.