Louis J. Gross

Applied Mathematical Ecology

I. Introduction.- Ecology in Theory and Application.- II. Resource Management.- Bioeconomic Modeling and Resource Management.- Common Property and the Conservation of Natural Resources.- Information and Area-Wide Control in Agricultural Ecology.- III. Epidemiology.- Fundamental Aspects of Epidemiology.- Three Basic Epidemiological Models.- The Population Biology of Parasitic Helminths in Animal Populations.- Simple Versus Complex Epidemiological Models.- Periodicity in Epidemiological Models.- Case Studies.- Rubella.- Influenza and Some Related Mathematical Models.- Review of Recent Models of HIV/AIDS Transmission.- The Transmission Dynamics of Human Immunodeficiency Virus (HIV).- IV. Ecotoxicology.- Models in Ecotoxicology: Methodological Aspects.- Deterministic and Statistical Models of Chemical Fate in Aquatic Systems.- Effects of Toxicants on Aquatic Populations.- V. Demography and Population Biology.- Mathematical Models in Plant Biology: An Overview.- Stable Population Theory and Applications.- Stage Structure Models Applied in Evolutionary Ecology.- Some Applications of Structured Models in Population Dynamics.- Author Index.

Landscape Modeling for Everglades Ecosystem Restoration

ABSTRACT A major environmental restoration effort is under way that will affect the Everglades and its neighboring ecosystems in southern Florida. Ecosystem and population-level modeling is being used to help in the planning and evaluation of this restoration. The specific objective of one of these modeling approaches, the Across Trophic Level System Simulation (ATLSS), is to predict the responses of a suite of higher trophic level species to several proposed alterations in Everglades hydrology. These include several species of wading birds, the snail kite, Cape Sable seaside sparrow, Florida panther, white-tailed deer, American alligator, and American crocodile. ATLSS is an ecosystem landscape-modeling approach and uses Geographic Information System (GIS) vegetation data and existing hydrology models for South Florida to provide the basic landscape for these species. A method of pseudotopography provides estimates of water depths through time at 28 × 28-m resolution across the landscape of southern Florida. Hydrologic model output drives models of habitat and prey availability for the higher trophic level species. Spatially explicit, individual-based computer models simulate these species. ATLSS simulations can compare the landscape dynamic spatial pattern of the species resulting from different proposed water management strategies. Here we compare the predicted effects of one possible change in water management in South Florida with the base case of no change. Preliminary model results predict substantial differences between these alternatives in some biotic spatial patterns.

Vegetation, fire, and feedbacks: a disturbance-mediated model of savannas.

Savanna models that are based on recurrent disturbances such as fire result in nonequilibrium savannas, but these models rarely incorporate vegetation feedbacks on fire frequency or include more than two states (grasses and trees). We develop a disturbance model that includes vegetation‐fire feedbacks, using a system of differential equations to represent three main components of savannas: grasses, fire‐tolerant savanna trees, and fire‐intolerant forest trees. We investigate the stability of savannas in the presence of positive feedbacks of fire frequency with (1) grasses, (2) savanna trees, and (3) grasses and savanna trees together while also allowing for negative feedbacks of forest trees on fire frequency. We find that positive feedbacks between fire frequency and savanna trees, alone or together with grasses, can stabilize savannas, blocking the conversion of savannas to forests. Negative feedbacks of forest trees on fire frequency shift the range of parameter space that supports savannas, but they do not generally alter our results. We propose that pyrogenic trees that modify characteristics of fire regimes are ecosystem engineers that facilitate the persistence of savannas, generating both threshold fire frequencies with rapid changes in community composition when these thresholds are crossed and hystereses with bistable community states.

Photosynthetic Dynamics in Varying Light Environments: A Model and Its Application to Whole Leaf Carbon Gain

The light environment of a plant microsite can be highly dynamic with sunflecks lasting from seconds to minutes in length. Whole leaf photosynthetic response to such variations is complex and species-specific. A mathematical model based on a single enzyme pool is derived to mimic leaf response in varying light. The model is used to analyze carbon gain in Fragaria virginiana. I find that sunflecks can be responsible for an appreciable proportion of leaf carbon gain throughout a day, especially under light-limited conditions. Model results also indicate that the standard methods of estimating carbon gain in varying light using steady-state photosynthetic rates often give overesti- mates.

A dynamic landscape model for fish in the Everglades and its application to restoration

Abstract A model (ALFISH) for fish functional groups in freshwater marshes of the greater Everglades area of southern Florida has been developed. Its main objective is to assess the spatial pattern of fish densities through time across freshwater marshes. This model has the capability of providing a dynamic measure of the spatially-explicit food resources available to wading birds. ALFISH simulates two functional groups, large and small fish, where the larger ones can prey on the small fish type. Both functional groups are size-structured. The marsh landscape is modeled as 500×500 m spatial cells on a grid across southern Florida. A hydrology model predicts water levels in the spatial cells on 5-day time steps. Fish populations spread across the marsh during flooded conditions and either retreat into refugia (alligator ponds), move to other spatial cells, or die if their cell dries out. ALFISH has been applied to the evaluation of alternative water regulation scenarios under the Central and South Florida Comprehensive Project Review Study. The objective of this Review Study is to compare alternative methods for restoring historical ecological conditions in southern Florida. ALFISH has provided information on which plans are most are likely to increase fish biomass and its availability to wading bird populations.

Landscape-based spatially explicit species index models for everglades restoration

As part of the effort to restore the ∼10 000-km2 Everglades drainage in southern Florida, USA, we developed spatially explicit species index (SESI) models of a number of species and species groups. In this paper we describe the methodology and results of three such models: those for the Cape Sable Seaside Sparrow and the Snail Kite, and the species group model of long-legged wading birds. SESI models are designed to produce relative comparisons of one management alternative to a base scenario or to another alternative. The model outputs do not provide an exact quantitative prediction of future biotic group responses, but rather, when applying the same input data and different hydrologic plans, the models provide the best available means to compare the relative response of the biotic groups. We compared four alternative hydrologic management scenarios to a base scenario (i.e., predicted conditions assuming that current water management practices continue). We ranked the results of the comparisons for each ...

The limits to prediction in ecological systems

Predicting the future trajectories of ecological systems is increasingly important as the magnitude of anthropogenic perturbation of the earth systems grows. We distinguish between two types of predictability: the intrinsic or theoretical predictability of a system and the realized predictability that is achieved using available models and parameterizations. We contend that there are strong limits on the intrinsic predictability of ecological systems that arise from inherent characteristics of biological systems. While the realized predictability of ecological systems can be limited by process and parameter misspecification or uncertainty, we argue that the intrinsic predictability of ecological systems is widely and strongly limited by computational irreducibility. When realized predictability is low relative to intrinsic predictability, prediction can be improved through improved model structure or specification of parameters. Computational irreducibility, however, asserts that future states of the system cannot be derived except through computation of all of the intervening states, imposing a strong limit on the intrinsic or theoretical predictability. We argue that ecological systems are likely to be computationally irreducible because of the difficulty of pre-stating the relevant features of ecological niches, the complexity of ecological systems and because the biosphere can enable its own novel system states or adjacent possible. We argue that computational irreducibility is likely to be pervasive and to impose strong limits on the potential for prediction in ecology.

Panthers and Forests in South Florida: an Ecological Perspective

The endangered Florida panther (Puma concolor coryi) survives in an area of pronounced habitat diversity in southern Florida, occupying extensive home ranges that encompass a mosaic of habitats. Twenty-one years of daytime monitoring via radiotelemetry have provided substantial but incomplete information about panther ecology, mainly because this method fails to capture movement and habitat use between dusk and dawn, when panthers are most active. Broad characterizations of panther habitat suitability have nonetheless been derived from telemetry-based habitat selection studies, focusing narrowly on forests where daytime resting sites are often located. The resulting forest-centered view of panthers attributed their restricted distribution and absence of population growth in the mid-1990s to a scarcity of unfragmented forest for expansion. However, the panther population has doubled since the beginning of genetic restoration in 1995, increasing five-fold in public areas described as unsuitable based on forest criteria. Although the forest-centered view no longer explains panther distribution, it continues to shape management decisions and habitat conservation policies. The assumptions and limitations of this view therefore merit critical examination. We analyze the role of forests in the ecology of the Florida panther. To address the absence of nighttime telemetry data, we use innovative telemetry mapping techniques and incorporate information from field observations indicating habitat use during active hours (e.g., tracks, scats, urine markers, and kill sites). We consider daytime telemetry data in the context of panther home ranges and breeding units. We analyze home range size in relation to the amount of forest within each range, concluding that percent forest cover is a poor predictor of size. We apply fractal analysis techniques to characterize the relative density of forest cover associated with daytime locations and interpret the results in terms of spatial landscape patterns, highlighting the limitations of daytime telemetry data for characterizing overall habitat use. We conclude that the forest-centered view of panther habitat selection is based on an uncritical evaluation of telemetry data collected prior to the recent population expansion and on the unsupported assumption that day bed habitats are representative of nighttime habitat use. We find that numerous factors contribute to habitat suitability and population density and distribution, and that P. concolor in Florida, as elsewhere in their range, are habitat generalists, exploiting the broad spectrum of available habitats for hunting, resting, mating, travel, denning, and dispersal. Whereas panthers readily use forested habitat with understory and prey, we find no support for the view that only the forested land within a habitat mosaic is potential panther habitat, or for the contention that only forested habitats are used by panthers within existing home ranges. We suggest a more ecologically consistent management and recovery paradigm based on maintaining the integrity of the system of overlapping home ranges that characterizes panther social structure and satisfies breeding requirements. Such a paradigm focuses on the requirements for reproductive success of a small population in a changing environment.

Periodic solutions to nonautonomous difference equations

A technique is presented for determining when periodic solutions to nonautonomous periodic difference equations exist. Under certain constraints, stable periodic solutions can be guaranteed to exist, and this is used to compare the analogous behavior of a nonautonomous periodic hyperbolic difference equation to that of the nonautonomous periodic Pearl-Verhulst logistic differential equation.

Testing the robustness of management decisions to uncertainty: Everglades restoration scenarios.

To effectively manage large natural reserves, resource managers must prepare for future contingencies while balancing the often conflicting priorities of different stakeholders. To deal with these issues, managers routinely employ models to project the response of ecosystems to different scenarios that represent alternative management plans or environmental forecasts. Scenario analysis is often used to rank such alternatives to aid the decision making process. However, model projections are subject to uncertainty in assumptions about model structure, parameter values, environmental inputs, and subcomponent interactions. We introduce an approach for testing the robustness of model-based management decisions to the uncertainty inherent in complex ecological models and their inputs. We use relative assessment to quantify the relative impacts of uncertainty on scenario ranking. To illustrate our approach we consider uncertainty in parameter values and uncertainty in input data, with specific examples drawn from the Florida Everglades restoration project. Our examples focus on two alternative 30-year hydrologic management plans that were ranked according to their overall impacts on wildlife habitat potential. We tested the assumption that varying the parameter settings and inputs of habitat index models does not change the rank order of the hydrologic plans. We compared the average projected index of habitat potential for four endemic species and two wading-bird guilds to rank the plans, accounting for variations in parameter settings and water level inputs associated with hypothetical future climates. Indices of habitat potential were based on projections from spatially explicit models that are closely tied to hydrology. For the American alligator, the rank order of the hydrologic plans was unaffected by substantial variation in model parameters. By contrast, simulated major shifts in water levels led to reversals in the ranks of the hydrologic plans in 24.1-30.6% of the projections for the wading bird guilds and several individual species. By exposing the differential effects of uncertainty, relative assessment can help resource managers assess the robustness of scenario choice in model-based policy decisions.