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The Lotka–Volterra predator-prey model makes a number of assumptions about the environment and biology of the predator and prey populations: [5] The prey population finds ample food at all times. The food supply of the predator population depends entirely on the size of the prey population.
A trophic function was first introduced in the differential equations of the Kolmogorov predator–prey model. It generalizes the linear case of predator–prey interaction firstly described by Volterra and Lotka in the Lotka–Volterra equation. A trophic function represents the consumption of prey assuming a given number of predators.
When the prey would become extinct locally at one habitat patch, they were able to reestablish by migrating to new patches before being attacked by predators. This habitat spatial structure of patches allowed for coexistence between the predator and prey species and promoted a stable population oscillation model. [6]
Predator–prey isoclines before and after pesticide application. Pest abundance has increased. Now, to account for the difference in the population dynamics of the predator and prey that occurs with the addition of pesticides, variable q is added to represent the per capita rate at which both species are killed by the pesticide. The original ...
Because the number of prey harvested by each predator decreases as predators become more dense, ratio-dependent predation is a way of incorporating predator intraspecific competition for food. Ratio-dependent predation may account for heterogeneity in large-scale natural systems in which predator efficiency decreases when prey is scarce. [1]
Huffaker was expanding upon Gause's experiments by further introducing heterogeneity. Gause's experiments had found that predator and prey populations would become extinct regardless of initial population size. However, Gause also concluded that a predator–prey community could be self-sustaining if there were refuges for the prey population.
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This model can be generalized to any number of species competing against each other. One can think of the populations and growth rates as vectors, α 's as a matrix.Then the equation for any species i becomes = (=) or, if the carrying capacity is pulled into the interaction matrix (this doesn't actually change the equations, only how the interaction matrix is defined), = (=) where N is the ...