Predator-prey problem

Let us now consider in detail the classical predator-prey problem, that is, the interaction between two wild-life species, the prey, which is a herbivore, and the predator, a carnivore. These two animals coinhabit a region where the prey have an abundant supply of natural vegetation for food, and the predators depend on the prey for their entire supply of food. This is a simplification of the real ecological system where more than two species coexist, and where predators usually feed on a variety of prey. The Lotka-Volterra equations have also been formulated for such... 

In order to formulate the predator-prey problem, we make the following assumptions ... 

Equations (l)-(3) constitute the complete mathematical formulation of the predator-prey problem based on assumptions (a) to (d). Different assumptions would yield another set of differential equations [see Problem (5.8)]. In addition, the choice of constants and initial conditions influence the solution of the differential equations and generate a diverse set of qualitative behavior patterns for the two populations. Depending on the form of the differential equations and the values of the constants chosen, the solution patterns may vary from stifle, damped oscillations, where the species reach their respective stable symbiotic population densities, to highly unstable situations, in which one of the species is driven to extinction while the other explodes to extreme population density. 

Although flie predator prey problem has a fairly simple set of nonlinear differential equations, it is typical of the nonlinear eoupling found in real world problems involving many sets of physieal parameters. 

Section 4.5). Of these, mesocosms have stimulated the greatest interest. In these, replicated and controlled tests can be carried out to establish the effects of chemicals upon the structure and function of the (artihcial) communities they contain. The major problem is relating effects produced in mesocosms to events in the real world (see Crossland 1994). Nevertheless, it can be argued that mesocosms do incorporate certain relationships (e.g., predator/prey) and processes (e.g., carbon cycle) that are found in the outside world, and they test the effects of chemicals on these. Once again, the judicious use of biomarker assays during the course of mesocosm studies may help to relate effects of chemicals measured by them with similar effects in the natural environment. 

The previous two chapters showed that competitive exclusion holds under a variety of conditions in the chemostat and its modifications. In this chapter it will be shown that if the competition is moved up one level - if the competition occurs among predators of an organism growing on the nutrient - then coexistence may occur. The fact that the competitors are at a higher trophic level allows for oscillations, and the coexistence that occurs is in the form of a stable limit cycle. Along the way it will be necessary to study a three-level food-chain problem which is of interest in its own right it is the chemostat version of predator-prey equations. The presentation follows that of [BHWl]. 

A variety of endpoints have been used to characterize the stress upon populations. Population numbers or density have been widely used for plant, animal, and microbial populations in spite of the problems in mark recapture and other sampling strategies. Since younger life stages are considered to be more sensitive to a variety of pollutants, shifts in age structure to an older population may indicate stress. Unfortunately, as populations mature, often age-structure comparisons become difficult. In addition, cycles in age structure and population size occur due to the inherent properties of the age structure of the population and predator-prey interactions. Crashes in populations such as that of the striped bass in the Chesapeake Bay do occur and certainly are observed. A crash often does not lend itself to an easy cause-effect relationship, making mitigation strategies difficult to create. 

As was mentioned in Subsection 5.6.2, stochastic versions of the Lotka-Volterra model lead to qualitatively different results from the deterministic model. The occurrence of a similar type of results is not too surprising. A simple model for random predator-prey interactions in a varying environment has been studied, staring from generalised Lotka-Volterra equations (De, 1984). The transition probability of extinction is to be determined. The standard procedure is to convert the problem to a Fokker-Planck equation (adopting continuous approximation) and to find an approximation procedure for evaluating the transition probabilities of extinction and of survival. 

Sometimes chemical problems can be answered using the knowledge from other sciences that are not related to chemistry at first sight. For example, some information about a complex reactions flow can be gained from the mathematical models of the interspecific competition. A classical example is the predator-prey model, which describes the population trends for predators and prey in living conditions... 

In terms of dealing with multi-objective problems, many different types of GA have been proposed, for example, vector-optimised evolution strategy, weight-based GA, niched-Pareto GA, predator-prey evolution strategy, Rudolph s elitist multi-objective evolutionary algorithm and distance-based Pareto AG. However, there is probably... 

Tliis problem focuses on using Polymath, an ordinary differential equation (ODE) solver, and also a non-linear equation (NLE) solver. Hiese equation solvers will be used extensively in later chapters. Infonnation on how to obtain and load the Polymath Software is given in Appendix E and on the DVD-ROM. (a) There are initially 400 rabbits (x) and 200 foxes (y) on Farmer Oat s property. Use Polymath or MATLAB to plot the concentration of foxes and rabbits as a function of time for a period of up to 600 days. The predator-prey relationships are given by the following set of coupled ordinary differential equations ... 

This section is meant to contribute to the old problem of the interaction between two biological species. To be or not to be is the essential question decided by the predator-prey interaction for the members of certain species. The famous Volterra-Lotka model for this problem [4.21,22] has attracted many researchers who have tried to generalize it in many respects [4.1, 6, 8, 9, 23-26]. 

As in the preceding section it is attempted to add a new aspect to an old problem In this case the new aspect is the combination of non-linear migration with the predator-prey interaction. It is useful to anticipate one main result of considering this combination ... 

Another interesting application is the use of evolving network models for a simulation approach to hard game-theoretical problems such as predator-prey models in continuous space-time. They cannot simply be reduced to discrete tournaments. One can find solutions to such problems by simulating the players as evolving neural nets [51]. In this model, the universal emergence of the main prey behavioral patterns observed in nature was verified in artificial evolution. The three stages are ... 

Predation (e.g. by foxes and birds of prey) may also be a serious cause of mortality in free-range husbandry. Foxes can be kept away with a solid fence, which can be supplemented by an electric fence. Birds of prey are not a big problem for poultry houses that are closed at night and free of holes. However, keeping birds of prey such as the hawks away is not easy, especially for farms surrounded by forest or in a landscape with a lot of trees. In such situations, low structures should be provided as hiding places for the hens. 

Disappearance of predators may also imbalance the equilibrium, and the problem scales up, such as the disappearance of foxes, predators of the deer mouse, which has allowed spreading of the hantavirus in the US, carried by mice (Levins 1993). Similarly, Sabia virus has emerged in Brazil, Guaranito virus in Venezuela (Lisieux 1994), machupo virus in Bolivia, and Junin fever in Argentine (Garrett 1994). In contrast, in a robust ecosystem, elimination of a predator provides space for another predator, such as in the disappearance of the coyote, which has opened the control of field mice to snakes and owls. When both predator and prey are endangered, it may occur that the prey develops resistance. This is taken into account in Volterra s equation (Ehrlich 1986). 

A biological interpretation of this problem in terms of population dynamics could be as follows both prey animals A and predators B living on them are reproduced by division in a medium with a spontaneous production of food E for them. With these studies in mind the model has to be slightly modified any new predator B is bom after killing the prey animal A with the probability < 1, which needs in our statement of the problem just a trivial replacement of /zB(r) for /ttB(r). 

Going back to D Ancona s problem of inhibited fishing in the Adria, we include the effects of fishing into the Lotka-Volterra model (15.1) by introducing a linear loss term with harvesting rate h for both predator and prey species... 

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