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A hybrid model, on the competition tumor cells immune system, is studied under suitable hypotheses. The explicit form for the equations is obtained in the case where the density function of transition is expressed as the product of separable functions. A concrete application is given starting from a modified Lotka-Volterra system of equations.

The competition between tumor cells and the immune system is mainly due to a significant presence of the proliferation and/or destructive events. In particular, cancer cells have the ability of expressing their biological activity to escape from the immune system which, in principle, have to challenge the progressing cells. The biological activity is not generally the same for all cells since it is statically distributed.

Several authors [

Other authors [

In some recent papers [

This time-depending stochastic parameter was linked [

In this paper, we study the above hybrid model by assuming a particular form of the stochastic coefficient. There follow interesting results on the model and, moreover, the classical model of Lotka-Volterra modified by the hiding-learning process can be derived as a special case.

Let us consider a system of two interacting and competing populations. Each population is constituted by a large number of individuals called active particles; their microscopic state is called (biological) activity. This activity enables the particle to organize a suitable response with respect to any information process. In absence of prior information, the activity reduces either to a minimal loss of energy or to a random process.

In active particle competitions, the simplest model of binary interaction is based on proliferation-destructive competition. So that, when the first population get aware of the existence of the other challenging population, it starts to proliferate and destroy the competing cells. However, in this process the most important step is the ability of cells to hide themselves and to learn about the activity of the competing population.

In details consider a physical system of two interacting populations each one constituted by a large number of active particles with sizes:

Particles are homogeneously distributed in space, while each population is characterized by a microscopic state, called activity, denoted by the variable

The description of the overall distribution over the microscopic state within each populations is given by the probability density function:

Moreover, it is

We will see in Sections

We consider, in this section, the competition between two cell populations: the first one with uncontrolled proliferating ability and with hiding ability; the second one with higher destructive ability, but with the need of learning about the presence of the first population. The analysis developed in what follows is referring to a specific case where the second population attempts to learn about the first population which, instead, escapes by modifying its appearance. Specifically, the hybrid evolution equations can be formally written as follows:

As a consequence, (

The derivation of (_{2} can be obtained starting from a detailed analysis of microscopic interactions. Specifically, consider binary interactions between a test, or candidate, particle with state

The encounter rate, which depends, for each pair of interacting populations on a suitable average of the relative velocity

The transition density function _{2}.

Since our model is based on the hiding-learning dynamics, one has to introduce the functional which takes into account the “distance” between the two distribution so that

In some recent papers [

In this case, it is

Thus, we have
_{1} to the microscopic model (_{2}.

In order to find some classes of solutions of (_{2} are given.

As an example, let us solve this system under the following hypotheses:
_{1}

It is well known that the pioneering Lotka-Volterra's model of two interacting and competing populations (

In this model, the hiding-learning processes are not considered and the interaction and competition of the two populations start immediately. The orbits of the solutions of (

Time evolutions of the orbits of (

If the hiding-learning processes occur, by using the results discussed in the previous sections, we propose the following system:

Time evolutions of the orbits of (

The system (

From Figure

In this paper, it has been studied a hybrid system of competition tumor cells versus immune system, within the kinetic model. A stochastic parameters is computed explicitly in the case of special transition density functions. A simple application shows that due to this parameters we obtain some more realistic solutions of the Lotka-Volterra system, where the cicle around the nonzero equilibrium point is shifted in time, thus showing the importance of the stochastic parameters in a correct approach to the analysis of competition models.