Stochastic population growth in spatially heterogeneous environments

Classical ecological theory predicts that environmental stochasticity increases extinction risk by reducing the average per-capita growth rate of populations. To understand the interactive effects of environmental stochasticity, spatial heterogeneity, and dispersal on population growth, we study the following model for population abundances in $n$ patches: the conditional law of $X_{t+dt}$ given $X_t=x$ is such that when $dt$ is small the conditional mean of $X_{t+dt}^i-X_t^i$ is approximately $[x^i\mu_i+\sum_j(x^j D_{ji}-x^i D_{ij})]dt$, where $X_t^i$ and $\mu_i$ are the abundance and per capita growth rate in the $i$-th patch respectivly, and $D_{ij}$ is the dispersal rate from the $i$-th to the $j$-th patch, and the conditional covariance of $X_{t+dt}^i-X_t^i$ and $X_{t+dt}^j-X_t^j$ is approximately $x^i x^j \sigma_{ij}dt$. We show for such a spatially extended population that if $S_t=(X_t^1+...+X_t^n)$ is the total population abundance, then $Y_t=X_t/S_t$, the vector of patch proportions, converges in law to a random vector $Y_\infty$ as $t\to\infty$, and the stochastic growth rate $\lim_{t\to\infty}t^{-1}\log S_t$ equals the space-time average per-capita growth rate \sum_i\mu_i\E[Y_\infty^i] experienced by the population minus half of the space-time average temporal variation \E[\sum_{i,j}\sigma_{ij}Y_\infty^i Y_\infty^j] experienced by the population. We derive analytic results for the law of $Y_\infty$, find which choice of the dispersal mechanism $D$ produces an optimal stochastic growth rate for a freely dispersing population, and investigate the effect on the stochastic growth rate of constraints on dispersal rates. Our results provide fundamental insights into "ideal free" movement in the face of uncertainty, the persistence of coupled sink populations, the evolution of dispersal rates, and the single large or several small (SLOSS) debate in conservation biology.Comment: 47 pages, 4 figure

Spatial and stochastic epidemics : theory, simulation and control

It is now widely acknowledged that spatial structure and hence the spatial position of host populations plays a vital role in the spread of infection. In this work I investigate an ensemble of techniques for understanding the stochastic dynamics of spatial and discrete epidemic processes, with especial consideration given to SIR disease dynamics for the Levins-type metapopulation. I present a toolbox of techniques for the modeller of spatial epidemics. The highlight results are a novel form of moment closure derived directly from a stochastic differential representation of the epidemic, a stochastic simulation algorithm that asymptotically in system size greatly out-performs existing simulation methods for the spatial epidemic and finally a method for tackling optimal vaccination scheduling problems for controlling the spread of an invasive pathogen

MODELING THE DYNAMICS OF INFECTIOUS DISEASES WITH Latency IN SPATIALLY HETEROGENEOUS Environments

Assuming that an infectious disease has a fixed latent period and latent individuals in the population may disperse in a spatially heterogeneous environment, we derive three new models of SIR type, which are more realistic than the existing related ones. The first one considers a 2-patch environment and ignores the demographic structure. The model is given by a system of delay differential equations (DDEs). It is a generalization of the classical Kermack-McKendrick SIR model, and it preserves some properties that the Kermack-McKendrick model processes. We show that the ratio of final sizes in two patches is fully determined by the ratio of dispersion rates of susceptible individuals between the two patches. We also numerically explore the patterns by which the disease dies out and have observed multiple outbreaks of the disease before it goes to extinction. The second model considers a general n-patch environment but incorporates a simple demographic structure, also resulting in a system of DDEs. Assuming the irreducibility of dispersal rates matrices of the infected classes, an expression of the basic reproduction number Ro is obtained. It is shown that disease free equilibrium is globally asymptotically stable if Ro \u3c 1, and unstable if Ro \u3e 1. In the latter case, there is at least one interior equilibrium and the disease is uniformly persistent. For n = 2, two special cases are considered to obtain more detailed results on how the disease latency and the population dispersal jointly affect the disease dynamics. The third model deals with a spatially continuous environment, and is given by a delayed system of reaction-diffusion equations with a spatially non-local term. We address the well-posedness of the model but the main concern is traveling wave fronts iii of the model. We obtain a critical value c* which is shown to be a lower bound for the wave speed of traveling wave fronts. Although we can not prove that this value is exactly the minimal wave speed, numeric simulations seem to suggest that it is. Furthermore, the simulations on the model equations also suggest that the disease spread speed coincides with c*. We also discuss how the model parameters affect c*

Two Biological Applications of Optimal Control to Hybrid Differential Equations and Elliptic Partial Differential Equations

In this dissertation, we investigate optimal control of hybrid differential equations and elliptic partial differential equations with two biological applications. We prove the existence of an optimal control for which the objective functional is maximized. The goal is to characterize the optimal control in terms of the solution of the opti- mality system. The optimality system consists of the state equations coupled with the adjoint equations. To obtain the optimality system we differentiate the objective functional with respect to the control. This process is applied to studying two prob- lems: one is a type of hybrid system involving ordinary differential equations and a discrete time feature. We apply our approach to a tick-transmitted disease model in which the tick dynamics changes seasonally while hosts have continuous dynam- ics. The goal is to maximize disease-free ticks and minimize infected ticks through an optimal control strategy of treatment with acaricide. The other is a semilinear elliptic partial differential equation model for fishery harvesting. We consider two objective functionals: maximizing the yield and minimizing the cost or variation in the fishing effort (control). Existence, necessary conditions and uniqueness for the optimal control for both problems are established. Numerical examples are given to illustrate the results

Demographic noise can reverse the direction of deterministic selection

Deterministic evolutionary theory robustly predicts that populations displaying altruistic behaviours will be driven to extinction by mutant cheats that absorb common benefits but do not themselves contribute. Here we show that when demographic stochasticity is accounted for, selection can in fact act in the reverse direction to that predicted deterministically, instead favouring cooperative behaviors that appreciably increase the carrying capacity of the population. Populations that exist in larger numbers experience a selective advantage by being more stochastically robust to invasions than smaller populations, and this advantage can persist even in the presence of reproductive costs. We investigate this general effect in the specific context of public goods production and find conditions for stochastic selection reversal leading to the success of public good producers. This insight, developed here analytically, is missed by both the deterministic analysis as well as standard game theoretic models that enforce a fixed population size. The effect is found to be amplified by space; in this scenario we find that selection reversal occurs within biologically reasonable parameter regimes for microbial populations. Beyond the public good problem, we formulate a general mathematical framework for models that may exhibit stochastic selection reversal. In this context, we describe a stochastic analogue to r-K theory, by which small populations can evolve to higher densities in the absence of disturbance.Comment: 25 pages, 12 figure

Protecting valuable resources using optimal control theory and feedback strategies for plant disease management

Mathematical models of tree diseases often have little to say about how to manage established epidemics. Models often show that it is too late for successful disease eradication, but few study what management could still be beneficial. This study focusses on finding effective control strategies for managing sudden oak death, a tree disease caused by Phytophthora ramorum. Sudden oak death is a devastating disease spreading through forests in California and southwestern Oregon. The disease is well established and eradication is no longer possible. The ongoing spread of sudden oak death is threatening high value tree resources, including national parks, and culturally and ecologically important species like tanoak. In this thesis we show how the allocation of limited resources for controlling sudden oak death can be optimised to protect these valuable trees. We use simple, approximate models of sudden oak death dynamics, to which we apply the mathematical framework of optimal control theory. Applying the optimised controls from the approximate model to a complex, spatial simulation model, we demonstrate that the framework finds effective strategies for protecting tanoak, whilst also conserving biodiversity. When applied to the problem of protecting Redwood National Park, which is under threat from a nearby outbreak of sudden oak death, the framework finds spatial strategies that balance protective barriers with control at the epidemic wavefront. Because of the number of variables in the system, computational and numerical limitations restrict the control optimisation to relatively simple approximate models. We show how a lack of accuracy in the approximate model can be accounted for by using model predictive control, from control systems engineering: an approach coupling feedback with optimal control theory. Continued surveillance of the complex system, and re-optimisation of the control strategy, ensures that the result remains close to optimal, and leads to highly effective disease management. In this thesis we show how the machinery of optimal control theory can inform plant disease management, protecting valuable resources from sudden oak death. Incorporating feedback into the application of the resulting strategies ensures control remains effective over long timescales, and is robust to uncertainties and stochasticity in the system. Local management of sudden oak death is still possible, and our results show how this can be achieved