Stochastic ecology

Differences between deterministic and stochastic ecological models have been emphasised many times, explicity or implicitly. The main qualitative deviation in the behaviour of ecodynamic models can be interpreted in terms of survival and extinction. 

Jernigan, R. W. Tsokos, C. P. (1980). Simulation of a nonlinear stochastic ecology model. Appl. Math, and Comp., 1, 9-25. 

Nonequilibrium and stochastic processes are common and essential for the apparent spatial and temporal patterns and processes found in ecological systems. 

Perceived stability in ecological systems frequently takes the form of metastability achieved through structural and functional redundancy incorporated in space and time. Patterns that appear stable at one scale may be due to nonequilibrium and stochastic processes occurring at adjacent hierarchies of scale. 

Microcosms do not have some of the characteristics of naturally synthesized ecological structures. Perhaps primary is that multispecies toxicity tests are by nature smaller in scale, thus reducing the number of species that can survive in these enclosed spaces compared to natural systems. This feature is very important since after dosing, every experimental design must make each replicate an island to prevent cross contamination and to protect the environment. Therefore the dynamics of extinction and the coupled stochastic and deterministic features of island biogeography produce effects that must be separated from that of the toxicant. Ensuring that each replicate is as similar as possible over the short term minimizes the differential effects of the enforced isolation, but eventually divergence occurs. 

Natural variability is a basic characteristic of stressors and ecological components as well as the factors that influence their distribution (e.g., weather patterns, nutrient availability). As noted by Suter (1990b), of all the contributions to uncertainty, stochasticity is the only one that can be acknowledged and described but not reduced. Natural variability is amenable to quantitative analyses, including Monte Carlo simulation and statistical uncertainty analysis (O Neill and Gardner 1979, O Neill et al. 1982). 

We now realize that ecological systems are complex systems, dependent upon spatial and temporal scales, and that they have stochastic elements. Also, we see that the mechanisms of evolution which are operating pose barriers to the use of analysis of variance and similar conventional tools for the... 

Nonlinear dynamics of complex processes is an active research field with large numbers of publications in basic research and broad applications from diverse fields of science. Nonlinear dynamics as manifested by deterministic and stochastic evolution models of complex behaviour has entered statistical physics, physical chemistry, biophysics, geophysics, astrophysics, theoretical ecology, semiconductor physics and -optics etc. This research has induced a new terminology in science connected with new questions, problems, solutions and methods. New scenarios have emerged for spatio-temporal structures in dynamical systems far from equilibrium. Their analysis and possible control are intriguing and challenging aspects of the current research. 

Second stage - optimization. In case of having incomplete information about the system "pesticide-environment" determinated mathematical methods of analysis are of little use. That is why, in the block of optimization of the model system, a dynamic stochastic model based on Bellman s method of dynamic programming, has been used. Markov process was taken as a mathematical model of the system (Hovard, 1964). The main goal of the optimization model is to find out the optimal value of X taking into account the ecological negative influence of pesticide. In... 

For a more extensive overview of descriptive mnltivariate stochastic techniques it is referred to Hair et al. (1995), Berthouex and Brown (2002), and Schabenberger and Pierce (2002). For scaling methods in aquatic ecology it is referred to Seurant and Strntton (2004). 

After a general introductory discussion of the stochastic model the two models will both be defined. Relevant, and ecologically meaningful, differences between the two models will be discussed at the end. 

What are the special ecological reasons that make it plausible to use a stochastic model ... 

Finally, let us mention an argument for the stochastic model coming from the endeavour to revitalise the science of ecological modelling. Large multiparameter systems aimed at a detailed description of lake eutrophication seem to have become less popular in recent years. This may either be the consequence of either the mathematical and computational problems involved or be due to the lack of measurements, or both. It is hoped that stochastic models constitute a possibility in this regard. 

Very few stochastic lake models seem to exist in the literature. Ecological problems related to those investigated here are described stochastically by j the following tools time series, random walk, diffusion processes, differential/ equations with random parameters (taken in the wide sense), and Markovia pure jump processes. 

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