Predator-prey model - Scholarpedia
the predator species is totally dependent on a single prey species as its only food Very few such "pure" predator-prey interactions have been observed in nature, . The Lotka-Volterra model consists of a system of linked differential equations that (Note: This is a general result in calculus -- not particular to this setting. Suppose those of age have fertility Then the equation below. The Lotka-Volterra equations describe an ecological predator-prey (or parasite- host) model which assumes that, for a set of fixed positive constants A.
Show that there is a pair xs, ys of nonzero equilibrium populations. Express xs and ys in terms of a, b, c, and p. This is a general result in calculus -- not particular to this setting. Give a justification for this result. Use the preceding step to write a single differential equation for y as a function of x, with the time variable t eliminated from the problem.Differential Equations, Lecture 8.3: Predator-prey models
The solutions of this differential equation are called trajectories of the system. Separate the variables and integrate to find an equation that defines the trajectories. Different values of the constant of integration give different trajectories. What do you learn about trajectories from the solution equation?
Step 2 also allows us to draw a direction field for trajectories.
Use the sample values for a, b, c, and p in your worksheet to construct a direction field. What do you learn about trajectories from the direction field? Add several trajectories to your direction field.
Do they agree with what you said about trajectories in the preceding step?
Calculate xs and ys for the sample values for a, b, c, and p in your worksheet. The entire term, ca'PN, tells us that increases in the predator population are proportional to the product of predator and prey abundance. Turning to the prey population, we would expect that without predation, the numbers of prey would increase exponentially.
The following equation describes the rate of increase of the prey population with respect to time, where r is the growth rate of the prey population, and N is the abundance of the prey population: Here the term a'PN reflects the fact that losses from the prey population due to predation are proportional to the product of predator and prey abundances.
Equations 2 and 4 describe predator and prey population dynamics in the presence of one another, and together make up the Lotka-Volterra predator-prey model. The model predicts a cyclical relationship between predator and prey numbers: Increase in consumption rate, however, has an obvious consequence-- a decrease in the number of prey Nwhich in turn causes P and therefore a'PN to decrease.
As a'PN decreases the prey population is able to recover, and N increases. Now P can increase, and the cycle begins again.
This graph shows the cyclical relationship predicted by the model for hypothetical predator and prey populations. He was compelled to resign his university post and membership of scientific academies, and, during the following years, he lived largely abroad, returning to Rome just before his death.
Kermack-McKendrick Model There is herd immunity in predation and in epidemics. It is convenient to frame this in terms of epidemiology where now we refer to the prey as being susceptibles and the predators as being infectives. Consider a time interval that is short compared to reproduction of the susceptible population, i. However, if this condition is not satisfied, the infective population will decrease.
Kermack-McKendrick model of propagation of infectious disease. This indicates the size of the susceptible population that avoids the infection.
Lotka-Volterra Equations -- from Wolfram MathWorld
Surprisingly, this number is always greater than zero, which shows that some susceptibles will always survive!
It reflects the fact that the susceptible population can be reduced to a level below which infectives will not increase. The model in this case is referred to as being the Kermack-McKendrick model of susceptible-infective interactions in epidemiology.
Jacob-Monod Model Another approach to modeling the interaction between prey and predators was developed to account as well for organisms such as bacteria taking up nutrients. There is a limited uptake rate that such organisms are capable of, and the next model accounts for limited uptake rates. This model was discovered independently in several diverse applications.
It is akin to the Haldane-Briggs model and Michaelis-Menten model in biochemistrythe Jacob-Monod model in microbial ecology, and the Beverton-Holt model in fisheries. It serves as one of the important building blocks in studies of complex biochemical reactions and in ecology Smith and Waltman A typical use of this model is to describe a continuous-flow growth device, such as a chemostatwhere there is continuous removal of nutrient and feeders and a continuous supply of fresh nutrient.
The Jacob-Monod model is used to describe such a bacterial growth device, for example to determine conditions for a sustained dynamic equilibrium to exist by balancing growth due to uptake of nutrient with wash out of feeders. Some predator-prey models use terms similar to those appearing in the Jacob-Monod model to describe the rate at which predators consume prey.
More generally, any of the data in the Lotka-Volterra model can be taken to depend on prey density as appropriate for the system being studied.