Stochastic population growth in spatially heterogeneous environments: The density-dependent case

This work is devoted to studying the dynamics of a structured population that is subject to the combined effects of environmental stochasticity, competition for resources, spatio-temporal heterogeneity and dispersal. The population is spread throughout $n$ patches whose population abundances are modelled as the solutions of a system of nonlinear stochastic differential equations living on $[0,\infty)^n$. We prove that $r$, the stochastic growth rate of the total population in the absence of competition, determines the long-term behaviour of the population. The parameter $r$ can be expressed as the Lyapunov exponent of an associated linearized system of stochastic differential equations. Detailed analysis shows that if $r>0$, the population abundances converge polynomially fast to a unique invariant probability measure on $(0,\infty)^n$, while when $r<0$, the population abundances of the patches converge almost surely to $0$ exponentially fast. This generalizes and extends the results of Evans et al (2014 J. Math. Biol.) and proves one of their conjectures. Compared to recent developments, our model incorporates very general density-dependent growth rates and competition terms. Furthermore, we prove that persistence is robust to small, possibly density dependent, perturbations of the growth rates, dispersal matrix and covariance matrix of the environmental noise. Our work allows the environmental noise driving our system to be degenerate. This is relevant from a biological point of view since, for example, the environments of the different patches can be perfectly correlated. As an example we fully analyze the two-patch case, $n=2$, and show that the stochastic growth rate is a decreasing function of the dispersion rate. In particular, coupling two sink patches can never yield persistence, in contrast to the results from the non-degenerate setting treated by Evans et al.Comment: 43 pages, 1 figure, edited according to the suggestion of the referees, to appear in Journal of Mathematical Biolog

Natural Variation and Neuromechanical Systems

Natural variation plays an important but subtle and often ignored role in neuromechanical systems. This is especially important when designing for living or hybrid systems \ud which involve a biological or self-assembling component. Accounting for natural variation can be accomplished by taking a population phenomics approach to modeling and analyzing such systems. I will advocate the position that noise in neuromechanical systems is partially represented by natural variation inherent in user physiology. Furthermore, this noise can be augmentative in systems that couple physiological systems with technology. There are several tools and approaches that can be borrowed from computational biology to characterize the populations of users as they interact with the technology. In addition to transplanted approaches, the potential of natural variation can be understood as having a range of effects on both the individual's physiology and function of the living/hybrid system over time. Finally, accounting for natural variation can be put to good use in human-machine system design, as three prescriptions for exploiting variation in design are proposed

Persistence in fluctuating environments

Understanding under what conditions interacting populations, whether they be plants, animals, or viral particles, coexist is a question of theoretical and practical importance in population biology. Both biotic interactions and environmental fluctuations are key factors that can facilitate or disrupt coexistence. To better understand this interplay between these deterministic and stochastic forces, we develop a mathematical theory extending the nonlinear theory of permanence for deterministic systems to stochastic difference and differential equations. Our condition for coexistence requires that there is a fixed set of weights associated with the interacting populations and this weighted combination of populations' invasion rates is positive for any (ergodic) stationary distribution associated with a subcollection of populations. Here, an invasion rate corresponds to an average per-capita growth rate along a stationary distribution. When this condition holds and there is sufficient noise in the system, we show that the populations approach a unique positive stationary distribution. Moreover, we show that our coexistence criterion is robust to small perturbations of the model functions. Using this theory, we illustrate that (i) environmental noise enhances or inhibits coexistence in communities with rock-paper-scissor dynamics depending on correlations between interspecific demographic rates, (ii) stochastic variation in mortality rates has no effect on the coexistence criteria for discrete-time Lotka-Volterra communities, and (iii) random forcing can promote genetic diversity in the presence of exploitative interactions.Comment: 25 page

Analysis of Noisy Evolutionary Optimization When Sampling Fails

In noisy evolutionary optimization, sampling is a common strategy to deal with noise. By the sampling strategy, the fitness of a solution is evaluated multiple times (called \emph{sample size}) independently, and its true fitness is then approximated by the average of these evaluations. Previous studies on sampling are mainly empirical. In this paper, we first investigate the effect of sample size from a theoretical perspective. By analyzing the (1+1)-EA on the noisy LeadingOnes problem, we show that as the sample size increases, the running time can reduce from exponential to polynomial, but then return to exponential. This suggests that a proper sample size is crucial in practice. Then, we investigate what strategies can work when sampling with any fixed sample size fails. By two illustrative examples, we prove that using parent or offspring populations can be better. Finally, we construct an artificial noisy example to show that when using neither sampling nor populations is effective, adaptive sampling (i.e., sampling with an adaptive sample size) can work. This, for the first time, provides a theoretical support for the use of adaptive sampling

Uncertainty And Evolutionary Optimization: A Novel Approach

Evolutionary algorithms (EA) have been widely accepted as efficient solvers for complex real world optimization problems, including engineering optimization. However, real world optimization problems often involve uncertain environment including noisy and/or dynamic environments, which pose major challenges to EA-based optimization. The presence of noise interferes with the evaluation and the selection process of EA, and thus adversely affects its performance. In addition, as presence of noise poses challenges to the evaluation of the fitness function, it may need to be estimated instead of being evaluated. Several existing approaches attempt to address this problem, such as introduction of diversity (hyper mutation, random immigrants, special operators) or incorporation of memory of the past (diploidy, case based memory). However, these approaches fail to adequately address the problem. In this paper we propose a Distributed Population Switching Evolutionary Algorithm (DPSEA) method that addresses optimization of functions with noisy fitness using a distributed population switching architecture, to simulate a distributed self-adaptive memory of the solution space. Local regression is used in the pseudo-populations to estimate the fitness. Successful applications to benchmark test problems ascertain the proposed method's superior performance in terms of both robustness and accuracy.Comment: In Proceedings of the The 9th IEEE Conference on Industrial Electronics and Applications (ICIEA 2014), IEEE Press, pp. 988-983, 201