Spatial Ecology: Patterns and Processes

Mathematical modelers hardly have sound knowledge of biological systems they intend to explore. Therefore, it is essential to introduce key concepts; e.g., populations, species, communities, etc. It presents a description of how diversity of species is organized in different taxonomic classes. All relevant phenomena which play important role in spatial systems are discussed. Allee effect is a phenomenon which governs the rate of growth of a population at low population densities. At higher population densities, growth of a population is limited by its carrying capacity. Habitat fragmentation and Allee effect are two key factors which determine the population growth and community structure. The chapter identifies challenges for a mathematical modeler in the present day scenario and indicates how these challenges could be handled in future. It also describes how the eBook is organized.

This chapter provides a detailed description of three basic population interactions in ecology: predation, competition and mutualism. The per capita rate at which a predator depletes population of its prey is known as functional response of the predator. The issue of prey–dependent vs. predator–dependent functional response is discussed in detail. The prey–dependent functional responses are presented in such a way that it becomes easy to work out how a given functional response is related to any other. How competitive exclusion principle work is demonstrated with the help of isoclines. A lucid description of relatively difficult and subtle interactions such as apparent competition and competitive displacement are presented. At the end, the chapter briefly describes two kinds of mutualistic interactions: Symbiotic mutualism and Non-symbiotic mutualism.

The chapter presents history of modeling in ecology briefly. Starting from initial efforts of A. J. Lotka and Vito Volterra, it discusses all the models of well–mixed type which are represented by difference or differential equations. Merits and demerits of every model is presented. Well–mixed mathematical models (WMMs) described by coupled ordinary differential equation represent ecosystems which are realized in “micro-cosm” experiments. A mathematical theory of ecological chaos is presented which makes testable predictions. Theory is based on mathematical models of ecological communities described by ordinary differential equations. It is shown that deterministic chaos exists in narrow parameter ranges. Non-linear dynamics (oscillations and chaos) favor species coexistence. It was demonstrated that population dynamics of species competing for abiotic resources could display oscillations and chaos. The model, that these investigators used, belong to a new class of models called resource competition models which link the population dynamics of competing species with the dynamics of the resources that these species are competing for. An attractive feature of these models is that they use biological traits of species to predict the dynamics of competition.

Understanding ecosystem’s response to perturbations is essential to get an idea of ecological organization and function. May started a debate in 1973. The debate revolved around ‘stability’ and ‘complexity’ in ecological systems. In the same year, C. S. Holling in British Columbia came up with imaginative theories of ecosystem function. Engineering resilience and ecological resilience are two main tenets of his theory. These theories help us analyze an ecosystem’s response to perturbations; e.g., changes in sea surface temperature (SST), other climatic variables, disease and habitat fragmentation, etc. Ecological systems are complex systems. An idiosyncratic feature of complex systems is that the whole behaves in an entirely different fashion than the parts. Concepts and techniques of Newtonian mechanics hardly apply to such systems. What applies to them is new kind of dynamics called non–linear dynamics. The Chapter describes all necessary concepts from this discipline. The research on discontinuities in ecological systems suggests the presence of adaptive cycles across the scales of a panarchy; a nested set of adaptive cycles operating at discrete levels. A system’s resilience depends on the interconnections between structure and dynamics at multiple scales. Complex systems are more resilient when the threshold between a given dynamic regime and an alternate regime is higher.

History of pattern formation dates back to 1952 when A. M. Turing pointed out that diffusion can destabilize an otherwise stable system to give rise to spatial patterns. Since then the instability has been studied in ecological, chemical and biochemical systems. An alternative to reaction–diffusion systems is meta–population models which assume that species can be thought as distributed in different spatial pockets connected by spatial processes such as migration and dispersal. Murdoch et al. (1992) explored the model proposed by Godfray and Pacala who assumed that within patch dynamics is described by Lotka–Volterra model. Spatial differences were created by making the prey birth rate in patch 2 (α2) greater than that in patch 1(α1). Prey moves symmetrically from one patch to the other; i.e., z1 = z2 . The meta-population is neutrally stable when birth rates of prey are equal. When significant difference in birth rates is created, oscillations in prey abundance in two patches become increasingly less correlated. This is associated with per capita prey immigration into a patch becoming increasingly temporally density–dependent. The density dependence arises as the number of immigrants into a patch is weakly correlated with the number of residents in the patch. Cellular automata simulation of a reaction–diffusion system obeying rules by Ebenhoh shows that fractals are present in fish school motion. Lewis and Collaborators developed a modeling approach which enables us to find out invasion speed of a biological invasion. This approach involves setting up an integro–differential equation which needs a dispersal kernel to be specified.

A theorem by A. N. Kolmogorov is presented which serves as a tool to identify dynamical regimes characterized by stable equilibria and sustained periodic oscillations. A valuable addition to Turing instability was the discovery of ‘ Wave of Chaos’ in 2001 by Petrovskii and Malchow who demonstrated that the phenomenon is common in ecological communities modeled by reaction–diffusion systems. Later authors attempted to incorporate directed animal movement called prey–taxis. The region around a Turing–Hopf bifurcation is vital for the formation of complex spatio– temporal patterns. Patchiness is manifested by spatial heterogeneity. A reaction– diffusion–advection model by Serizava and collaborators help simulate effect of eutrophication on aquatic systems as the nutrient level in the system is increased. A model by Lafferty and Holt clarifies dynamics of infection under environmental stress (toxic chemicals in pollution, malnutrition and thermal stress from climate change). The dynamics is represented by a system of ordinary differential equations. Simulation experiments on this model suggested that host–specific diseases declined with stress while non–specific diseases increased with stress.

Predation, herbivory, competition and mutualism are interactions which determine structure of biological communities in an ecosystem. A program in ecosystem studies consists of investigation of certain processes that link the living (or biotic) with the non–living (or abiotic). Energy transformation and biogeochemical cycle are two main ecosystem processes. The principle that spatial patterns affect ecological processes is the foundation of Landscape Ecology. Developmental activities cause habitat fragmentation. Two main effects of habitat fragmentation are: 1) the formation of isolated patches and 2) the increase in significance of ‘edge effects’. A theory which links extinction debt with habitat destruction in meta–population models is presented. Species Ability to Forestall Extinction (SAFE), has been developed to help those actively involved in conservation efforts. It is built on the concept of minimum population size required by a species to survive in the wild, known as minimum viable population (MVP) size in the literature, and measures how close species are to their minimum viable population sizes. SAFE measures distance between current abundance of a population from its minimum viable population size. It should be noted that SAFE is a probabilistic measure.