Spatial control is critical for optimal management of biophysical resources in ecological systems. At the same time, many biological resources of conservation interest are subject to random effects that pose substantial risks. Chance constraints in spatial mathematical programming models provide one useful way to integrate risks into plans for optimal management. A numerical estimation method is considered that appears helpful for heuristically solving these complex programming models. A hypothetical habitat resoration problem is presented.
A great variety of very practical issues in natural resource management involve spatial aspects of natural systems. Indeed, one of the most commonly applied computational tools by managers are geographic information systems (GIS) which provide a spatial view of data and the potential implications of management actions. Despite their prominence, GIS have very limited capability for either the dynamic modeling familiar to mathematical ecologists, or for linkage with optimization methods for dynamic control. I will start by discussing invasive species management from a very simple spatially-implicit model, expand this to a more realistic model for spatial control of invasive plants with application Lygodium macrophyllum in south Florida, mention some spatial control aspects of individual-based models, and end with some lessons learned from a long-term, complicated modeling project for Everglades restoration.
Simplified ordinary differential equation models of ecological systems have provided most of our theoretical understanding of consumer-resource dynamics. These models represent a macroscopic "mean field'' perspective that usually lumps together individuals differing in size, stage, mobility, physiological condition, genetic identity, micro-environment and many other details. One payoff of this simplification is a high degree of analytical and numerical tractability. Another is a clear experimental path to estimating regulatory mechanisms such as functional response curves. Functional responses are used in ODE models to estimate mean trophic rates as functions of mean resource and consumer densities. However, in most ecological systems resources and consumers are very heterogeneous in time and space. In heterogeneous landscapes, a given quantity of resource can be distributed in many ways, some of which result in higher consumption rates than others by specific types of consumers. This implies that functional responses cannot be functions only of mean resource and consumer densities. They may nonetheless be functions of mean densities along with a small number of other parameters. In this talk I will present a dimensional analysis that suggests what the other parameters might be, and how they might be used to derive ODE approximations for mobile consumers of heterogeneous resources using effective functional response curves.
Biological invasions often have dramatic ecological and economic consequences. Thus, there is keen interest in models that correctly predict rates of spread of invading organisms. In this talk, I discuss the formulation and analysis of integrodifference equations, link deterministic integrodifference equations to stochastic branching random walks, and show how these models shed light on the rate of spread of invading organisms.
Integrodifference equations are models that are discrete in time and continous in space. These equations model populations with discrete generations with separate growth and dispersal stages. The dispersal is modeled by an integral of the population density (after the growth) against a kernel. Optimal control of such hybrid systems is a new area and involves a combination of the techniques from the discrete version of Pontryagin's Maximum Principle and from control of partial differential equations. Analysis, characterizations of optimal controls and numerical illustrations will be given for some population examples.
There is a long history in the use of forest insect populations as model systems in the study of animal population dynamics. In this talk, I will provide an overview of how we have extended these studies to explore the spatial dynamics of forest insect populations. Much of this work has been motivated by the availability of digital maps that document the geographical extent of outbreaks of several forest insect species over successive years over large geographical regions. While the most striking temporal pattern evident in these data is the existence of periodicity in the presence of regional outbreaks, the most striking characteristic of the spatial dynamics of virtually all species is spatial synchrony. The term spatial synchrony refers to coincident changes in abundance among geographically disjunct populations. The ubiquitous presence of spatial synchrony provides an enticing challenge for population ecologists because this behavior may be caused by several different types of processes, most notably by a small amount of dispersal among populations or by the impact of a small but synchronous random effect, such as variation in weather. By comparing patterns of spatial synchrony among various species with varying dispersal capabilities, we have concluded that regional stochastic effects are the most likely cause of the ubiquitous synchrony in dynamics. However there is also evidence that long-distance dispersal can also greatly impact patterns of spatial dynamics as well. For example, populations of the larch budmoth in the European Alps exhibit recurring outbreak waves that move from west to east. We feel that the most likely cause of these population waves is an interaction between habitat heterogeneity (landscape connectivity) and the dominant reaction-diffusion processes that affect populations. We have also investigated how habitat heterogeneity can impact the synchronizing affect of regional stochasticity. Specifically, geographical variation in density-dependent population processes (caused by variation in habitat quality and other habitat characters) can greatly dilute the synchronizing effect of regional stochasticity. Geographical variation in habitat quality has probably received too little attention because it is one of the major determinants of observed patterns of spatial dynamics.
Joint work with Ottar N. Bjørnstad and Derek M. Johnson.
The question how populations in rivers can persist despite flow- induced washout has been termed the "drift paradox". More generally, systems with unidirectional flow and flow-induced wash-out include rivers, plug-flow reactors, prevailing wind directions, and climate- change models.
A first simple model in the form of a reaction-advection-diffusion equation explored persistence criteria by looking at the minimal domain size (Speirs and Gurney (2001), Ecology). Starting from this simple model, I will report on several extensions, namely: vertical structure in the population (drift and benthic state), spatial heterogeneity and the influence on channel geometry, effects of resource gradients, and competition of two species. I will focus on the minimal domain size, on speeds of upstream invasions, and on spatially mediated coexistence.
Most analyses of spatial fisheries models assume a single owner whose goal is the maximization of sustainable yield. These analyses ignore the redistribution of fishing effort in response to economics and regulation. We will describe a simple, spatial, bioeconomic model that accounts for the open-access nature of most marine fisheries. We have used the model to find the maximum sustainable economic rent that can be obtained using various policy instruments (including taxes on aggregate effort, taxes on aggregate catch, effort quotas and catch quotas). We contrast these solutions to the rent-maximizing distribution of effort employed by a sole owner and to the distribution of effort in unregulated open access (when all profits are dissipated). In many cases, the solution contains unexploited regions in space. The locations of the unexploited regions, and the potential sustainable rent that results, depends upon the policy instrument employed.
Optimal foraging and habitat selection theories that are based on non-spatial, deterministic models predict evolution towards generalist strategies in fine-grained habitats and towards specialization in coarse-grained habitats. In addition, coevolutionary processes appear to favor extreme specialization among parasites. We introduce a spatially explicit, stochastic model that confirms the effect of habitat coarseness on specialization in the absence of coevolutionary processes. To understand the effects of coevolutionary processes, we introduce feedback between hosts and their symbionts into our spatially explicit, stochastic model. We find that mutualists modify their habitat so that it becomes coarse-grained, and parasites modify their habitat so that it becomes fine-grained, suggesting that the lifestyle of the symbiont prevents habitat types from becoming extreme. This is joint work with Nicolas Lanchier, University of Minnesota.
It is well known that both space and stochasticity can play central roles in ecological systems. Theoretical ecologists have developed numerous approaches that apply to spatial and stochastic systems, such as simulations, pair-approximations, and spatial moment equations. However, these approaches are heuristic in the sense that they do not give a mathematically rigorous description of the system. For example, the usage of spatial moment equations involves a choice of moment closure, different choices leading to different answers.
We have developed a new method for the analysis of continuous-space continuous-time stochastic and spatial systems that is based on a systematic perturbation expansion of the underlying stochastic differential equations. The method allows one to analyze the spatial and stochastic model in an asymptotically (as interaction range tends to infinity) exact manner, in principle up to any order. Comparison with simulations show that the results are not only asymptotically correct but often good also when interactions are due to a few interacting neighbours only.
As an example, we apply the method to study (i) metapopulation dynamics in a correlated and dynamic landscape, (ii) the effects of habitat loss and fragmentation, and (iii) the effects of space and stochasticity on a community of competing plant species.
Range expansions of invading species in homogeneous environments have been extensively studied since the pioneer works by Fisher (1937) and Skellam (1951). However, environments for living organisms are often fragmented by natural or artificial habitat destruction.
Here we focus on how such environmental fragmentation affects the range expansion of invading species. We consider a single-species invasion in heterogeneous environments that are generated by segmenting an original favorable habitat into regularly striped, island-like, corridor-like, or randomly patched pattern. To deal with range expansion in such fragmented environments, we modify Fisher's equation by assuming that the intrinsic growth rate and diffusion coefficient vary depending on habitat properties.
By examining the traveling periodic wave (TPW) speed in the striped environment, we first derive the ray speed in a parametric form, from which the envelope of the expanding range can be predicted. The envelopes show varieties of patterns, nearly circular, oval-like, spindle-like or vanishing in the extreme case, depending on parameter values. By deriving the formula for the ray speed, we discuss how the pattern and speed of the range expansion are affected by the size of fragmentation, and the qualities of favorable and unfavorable habitats.
Secondly, we numerically solve extended Fisher's equation for island-like, corridor-like, and randomly patched environments with an initial distribution localized at the origin. The model is analyzed to examine how the spread of organisms is influenced by the patterns of habitat fragmentation, and which type of fragmentation is more favorable for species survival.