• Journal of Environmental Science International
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• v.20 no.2
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• pp.155-161
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• 2011

In application and discussion of population structure and phenotypic divergence in plant community, the classic Lotka-Volterra models of competition and spatial model are conceived as a mechanism that is composed by multiple interacting processes. Both the Lotka-Volterra and spatial simulation formulae predict that species diversity increases with genotypic richness (GR). The two formulae are also in agreement that species diversity generally decreases within increasing niche breadth (NB) and increases with increasing potential genotypic range (PGR). Across the entire parameter space in the Lotka-Volterra model and most of the parameter space in the spatial simulations, variance in community composition decreased with increasing genotypic richness. This was, in large part, a consequence of selecting genotypes randomly from a set pool.

• Journal for History of Mathematics
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• v.33 no.3
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• pp.167-177
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• 2020

Cooperative behavior may seem contrary to the notion of natural selection and adaptation, but is widely observed in nature, from the genetic level to the organism. The origin and persistence of cooperative behavior has long been a mystery to scientists studying evolution and ecology. One of the important research topics in the field of evolutionary ecology and behavioral ecology is to find out why cooperation is maintained over time. In this paper we take a historical overview of mathematical models representing cooperative relationships from the perspective of mathematical biology, which studies population dynamics between interacting biological groups, and analyze the mathematical characteristics and meanings of these cooperative models.

• Korean Journal of Mathematics
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• v.32 no.3
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• pp.381-406
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• 2024

The logistic growth model was developed with a single population in mind. We now analyze the growth of two interdependent populations, moving beyond the one-dimensional model. Interdependence between two species of animals can arise when one (the "prey") acts as a food supply for the other (the "predator"). Predator-prey models are the name given to models of this type. While social scientists are mostly concerned in human communities (where dependency hopefully takes various forms), predator-prey models are interesting for a variety of reasons. Some variations of this model produce limit cycles, an interesting sort of equilibrium that can be found in dynamical systems with two (or more) dimensions. In terms of substance, predator-prey models have a number of beneficial social science applications when the state variables are reinterpreted. This paper provides a quick overview of numerous predator-prey models with various types of behaviours that can be applied to ecological systems, based on a survey of various types of research publications published in the last ten years. The primary source for learning about predator-prey models used in ecological systems is historical research undertaken in various circumstances by various researchers. The review aids in the search for literature that investigates the impact of various parameters on ecological systems. There are also comparisons with traditional models, and the results are double-checked. It can be seen that several older predator-prey models, such as the Beddington-DeAngelis predator-prey model, the stage-structured predator-prey model, and the Lotka-Volterra predator-prey model, are stable and popular among academics. For each of these scenarios, the results are thoroughly checked.

• Journal for History of Mathematics
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• v.26 no.2_3
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• pp.197-210
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• 2013

The late 18C Malthus studied population growth for the first time, Verhulst the logistic model in 19C and, after that, the study of the predation competition between two species resulted in the appearance of Lotka-Volterra model and modified model supported by Gause's experiment with bacteria. Instable coexistence equilibrium being found, Solomon and Holling proposed functional and numerical response considering limited abilities of predator on prey, which applied to Lotka Volterra model. Nicholson and Baily, considering the predation between host and parasitoid in discrete time, made a model. In 20C there were developed various models of disease dynamics with the help of mathematics and real data and named SIS, SIR or SEIR on the basis of dynamical phenomena.

• Journal for History of Mathematics
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• v.25 no.1
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• pp.45-56
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• 2012

Mathematical biology has been recognized its importance recently and widely studied in the fields of mathematics, biology, medical sciences, and immunology. Mathematical ecology is an academic field that studies how populations of biological species change as times flows at specific locations in their habitats. It was the earliest form of the research field of mathematical biology and has been providing its basis. This article deals with various form of interactions between two biological species in a common habitat. Mathematical models of predator-prey type, competitive type, and simbiotic type are investigated.

• Korean Journal of Ecology and Environment
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• v.37 no.3 s.108
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• pp.304-312
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• 2004

The transient dynamics of three-trophic populations (prey, predator, and super predator) using ratio-dependent models responding to environmental impacts is analyzed. Environmental factors were divided into two parts: periodic factor (e.g., temperature) and general noise. Periodic factor was addressed as a frequency and bias, while general noise was expressed as a Gaussian distribution. Temperature bias ${\varepsilon}$, temperature frequency ${\Omega}$, and Gaussian noise amplitude ${\`{O}}$ accordingly revealed diverse status of population dynamics in three-trophic food chain, including extinction of species. The model showed stable limit cycles and strange attractors in the long-time behavior depending upon various values of the parameters. The dynamic behavior of the system appeared to be sensitive to changes in environmental input. The parameters of environmental input play an important role in determining extinction time of super predator and predator populations.