Lotka-Volterra competition model

The term -pj -pY in the kinetics of (8.8) represents the decrease in the cell division rate due to crowding. The constant y (> 1) expresses the proliferative advantage of the tumor cell population. Note that the model is in one-dimensional space and that the kinetic terms are of Lotka-Volterra competition type. 

The study of mathematical models of competition has led to the discovery of some very beautiful mathematics. This mathematics, often referred to as monotone dynamical systems theory, was largely developed by M. W. Hirsch [Hil Hi3], although others have made substantial contributions as well. In this section we describe a result that was first obtained in a now classical paper of DeMottoni and Schiaffino [DS] for the special case of periodic Lotka-Volterra systems. Later, it was recognized by Hale and Somolinos [HaS] and Smith [S4 S5] that the arguments in [DS] hold for general competitive and cooperative planar periodic systems. The result says that every bounded solution of such a system converges to a periodic solution that has the same period as the differential equation. 

In the next few sections we ll consider some simple examples of phase plane analysis. We begin with the classic Lotka-Volterra model of competition between two species, here imagined to be rabbits and sheep. Suppose that both species are competing for the same food supply (grass) and the amount available is limited. Furthermore, ignore all other complications, like predators, seasonal effects, and other sources of food. Then there are two main effects we should consider ... 

In the Lotka-Volterra model (3.68)-(3.69), f(P) = P. Model (3.71)-(3.72) is by no means the only possibility. Another popular choice is to assume that the functional response / depends not only on the prey, but it is ratio-dependent, i.e. it depends on the a mount of prey per predator f = f(P/Z), modelling competition for food among predators, that is absent in the previous models. An example is ... 

Competition among two species means that the increase in one of the populations decreases the net growth rate of the second one, and vice versa. This happens when they feed on the same resources, or if they produce substances (toxins) that are toxic for the other species. A classical competition model was also introduced in Volterra (1926), and considered in a more general parameter range by Lotka (1932). It is known as the competitive Lotka-Volterra system ... 

S. Pigolotti, C. Lopez, and E. Hernandez-Garcfa. Species clustering in competitive Lotka-Volterra models. Phys. Rev. Lett., 98 258101(1-4), 2007. 

Waltman of the University of Iowa recently pointed out to me in a personal communication that even when Lotka-Volterra concepts are discarded entirely and Monod s model is used for all growth rates, the resulting competition equations for two predators and one prey seem to have limit cycle solutions for certain conditions of operation. Mr. Basil Baltzls has found that use of a so-called multiple saturation model for the predators, which seems to be more appropriate than Monod s model for protozoans at any rate... 

The examination of competitive interactions among different species has been one of the main topics of mathematical biology. The most often used mathematical model is still a generalisation of the Lotka-Volterra model systems of polynomial ordinary differential equations expressible in terms of formal chemical reactions have also been investigated. The main problem is to find criteria for the coexistence of species. All species in the communities... 

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