A kinetic model with time‐dependent proliferative/destructive rates

This paper presents a new kinetic model with time‐dependent proliferative/destructive parameters, where the activity variable of the system attains its values in a discrete real subset. Therefore, a system of nonautonomous nonlinear ordinary differential equations is gained, with the related Cauchy problem. A first result of local existence and uniqueness of positive and bounded solution is proved. Then, the possibility of extend this result globally in time is discussed, with respect to the shape of nonconservative time‐dependent parameters. Numerical simulations are performed for some scenarios, corresponding to different shapes of time‐dependent parameters themselves. Furthermore, the pattern and long‐time behavior of solutions are numerically analyzed. Finally, these results show that equilibria and oscillations may occur.


INTRODUCTION
In an increasingly interconnected and complex world, the interest towards interacting systems has been widely growing in the last years.This interest leads to an interdisciplinarity, where the role of mathematics is reinforced thanks to its models and the increased availability of computing tools.The kinetic theory is one of the main mathematical tools in order to deal with modeling of an interacting systems (see [1,2] and references therein for a broader historical overview).For the aim of the current paper, we are interested in kinetic theory models of living systems (see [3,4] and references therein), where the whole system is composed of agents, generally called particles.The interactions between pairs of particles follow a binary and stochastic scheme according to the particular applications taken into account.Therefore, the shape of the interacting parameters is crucial for the description of the particular system.
The applications of the kinetic approach cover several sectors.Mathematical biology [3,5], medicine [6,7], vehicular traffic [8,9], and economy [10,11] are the best known examples, thanks to a long-time interest from several research groups.Recently, this interdisciplinarity of mathematics has also reached typical human sciences.For instance, there are opinion dynamics [12,13], psychology [14], human feelings [15], and the spreading of fake news [16].Moreover, the application in the context of energy and resource problem has also been considered in this framework [17].The recent COVID-19 pandemic event has determined an impact of these models on mathematical epidemiology [18,19].Finally, the behavior of people with respect to restriction measures had a significant impact on behavioral epidemiology [20].Also, in this field, the interdisciplinarity through kinetic theory provides some new results [21].
The kinetic description of a complex interacting system follows a multiscale structure (see [22][23][24] and references therein for details of multiscale modeling).A system of particles, also called agents, is considered.First, the system is divided into functional subsystem [25], such that particles/agents belonging to the same functional subsystem share the same strategy.The meaning of this strategy depends on the particular application, for instance, the wealth in a socio-economic system or the viral load in an epidemiological framework.The microscopic level of this description is defined by the stochastic binary interactions between particles, which depend on some parameters of the system.Then, the shape of these parameters strictly depends on the particular application taken into account.Specifically, the microscopic state is defined by one or more variables, that can be mechanical (i.e., space and velocity) or of other type.In the latter case, the microscopic variable is generally known as activity.Depending on the shape of the activity variable, the evolution of the system at the mesoscopic scale is expressed by integro-differential equations, partial-differential equations, or ordinary differential equations.As well as in the case of the functional subsystems, the meaning of the non-mechanical microscopic variables depends on the context under consideration.For instance, in a socio-economic system, functional subsystems represent the wealth levels, whereas the activity variable stands for the educational level of each agent.
In this paper, only a scalar non-mechanical activity real variable, attaining its values in a discrete subset of R, is considered.The reason for choosing a discrete microscopic variable u is twofold.On one hand, it is not restrictive with respect to applications.For instance, in a socio-economic system, the functional subsystems can represent the wealth classes, whereas, the activity can represent the educational level.In order to model this latter aspect, a discrete variable seems to be more convenient.On the other, this choice allows to gain some analytical and numerical properties improving the mathematical consistency of the model.The introduction of a suitable distribution function on each functional subsystem then furnishes the mesoscopic description of the system.Roughly speaking, this may also be indicated as a "statistical level."Then, the evolution of the system at mesoscopic level is gained by a system of nonlinear ordinary differential equations, of the first order.Assigning a suitable initial data, the related Cauchy problem is defined.Finally, the macroscopic description of the system in terms of global quantities is obtained by introducing some moments of distribution functions with respect to the activity variable.
The evolution of a stochastically interacting system, composed of particles, depends, among the others, on the parameters that define the microscopic interactions.This paper aims at analytically and numerically discussing a kinetic framework where not only conservative interactions occur, but still nonconservative ones are taken into account.This is not only a theoretical curiosity (still respectful in a research aim), but a dynamics of great interest when dealing with applications.For instance, if we want to describe the evolution of an ecological system, composed of two or more populations, birth/death rates cannot be neglected in order to have a more realistic description by the appropriate kinetic framework.The conservative interactions between pairs of individuals of the populations alone are not enough.Then, the introduction of nonconservative parameters is mandatory.In this paper, proliferative/destructive parameters are introduced in the evolution of the considered system.Generally, the introduction of nonconservative terms in a kinetic framework may cause some analytical problems.For instance, the positivity and boundedness of solution could not be assured.Indeed, blow-up phenomena may occur in these situations [26,27].Therefore, some further assumptions are required in order to preserve the typical properties of solutions of kinetic conservative models.
The kinetic framework proposed in this paper arises in this context.The main novelty is the time-dependent structure of parameters that characterize nonconservative interactions.As before, it is worth stressing out that this is not only a theoretical aspect.For instance, in an ecological system, the reproduction rate depends on the seasons [28], then an independent-time shape for the nonconservative parameters would not be realistic in that case.Also, when dealing with the immune system interactions, both the proliferation and the destruction of cells are relevant effects in the description that should not be neglected [29,30].Therefore, this new nonconservative framework can be applied in different contexts, beyond those already mentioned; these generality and versatility of the model represent one of the key highlights of the current work.Due to these considerations, we do not specialize the model with respect to a particular application, neither for analytical results, nor for numerical simulations, but we aim at further stressing its generality and versatility.Nevertheless, the analytical structure requires the analysis of existence and uniqueness of solution of the related Cauchy problem.In particular, the main problem of this framework still remains the study of boundedness of solution.However, some possible conditions for boundedness of solution are discussed, at least from a numerical viewpoint.Indeed, in the numerical simulations, a generic stochastically interacting system composed of two populations is modeled: Various scenarios are discussed with respect to different choices of the analytical shapes of time-dependent nonconservative parameters.
After this brief introduction, the contents of this paper are organized as follows, with four further sections.In Section 2, the new kinetic model with time-dependent proliferative/destructive parameters is developed, and the related Cauchy problem is formulated.The analytic results are discussed and presented in Section 3.After the proof of local existence and uniqueness, some considerations towards global existence in time are provided; in particular, the dependence with respect to the functional shape of the nonconservative parameters is analyzed.Then, in Section 4, some numerical simulations, applied in a general context with two interacting populations, are performed towards the new kinetic framework, by using standard MATLAB tools for ordinary differential equations, that is, a fourth-order Runge-Kutta method Specifically, five subsections investigate five different scenarios, in order to compare the behavior of the related solutions.Finally, the conclusions are stated, and possible future research perspectives are discussed in Section 5.

A NEW KINETIC MODEL
Let consider a many-particle system  composed of particles that have a stochastic interaction.Specifically, the system is divided into n ∈ N functional subsystems such that particles belonging to the same functional subsystem share the same strategy.The microscopic state of the system is described by a discrete variable u that acquires its values in a finite real subset  = {u 1 , u 2 , … , u n }, such that u i denotes the strategy or activity of the subsystem i, for i = 1, 2, … , n.
The particles of the system represent the agents of a particular dynamic, whose meaning depends on the specific application taken into consideration.
The distribution function of the ith functional subsystem is such that  i (t) gives the relative amount of particles of the ith functional subsystem at time t > 0, whereas f(t) = ( 1 (t),  2 (t), … ,  n (t)) is the vector distribution function of the overall system.For each p ∈ N 0 , the pth-order moment of the system with respect to u is By acquiring a physics viewpoint, the zeroth-order moment, the first-order moment, and the second-order moment correspond to density, linear u-momentum, and global activation energy, respectively.The interactions among the particles of the system are defined by the following quantities: • The interaction rate  hk , for h, k ∈ {1, 2, … , n}, that gives the number of encounters between particles of the hth function subsystem and particles of the kth functional subsystem; • The transition probability B i hk , for i, h, k ∈ {1, 2, … , n}, that gives the probability that a particle of the hth functional subsystem falls into the ith functional subsystem after interacting with a particle of the kth functional subsystem.Since B i hk is a probability, it is assumed that Then, the evolution of the ith functional subsystem is described by the following nonlinear ordinary differential equation: Specifically, is the gain term that gives the total number of particles that fall into the ith functional subsystem, after interacting with a particle of the kth functional subsystem; is the loss term that gives the number of particles that leave the ith functional subsystem due to the interaction with particles of the kth functional subsystem.
Then, the operator gives the net flux of particles related to the ith functional subsystem, for i ∈ {1, 2, … , n}.The framework (2) is globally conservative, in the sense that Therefore, we assume that the f is a probability, in the sense that Let us now consider that nonconservative events occur during the time-evolution of the system.Specifically, the following parameters are introduced: The functions  hk (t) represent the nonconservative time-dependent rates of proliferative or destructive type of particles of the ith functional subsystem due to interactions among particles of the hth functional subsystem and particles of the kth functional subsystem.In particular, if  hk (t) < 0, the interaction is destructive, whereas if  hk (t) > 0, the interaction is proliferative.
Bearing the conservative framework (2) in mind, the nonconservative kinetic framework is given by a system of nonlinear ordinary differential equations, derived from the conservative system by adding a new term associated with the nonlinear nonconservative interactions.Specifically, the evolution of the ith functional subsystem, for i ∈ {1, 2, … , n}, reads The operator P i [f](t), for i ∈ {1, 2, … , n}, is the nonconservative operator of the system.The system (3) rewrites, for Remark 1.If  hk (t) = 0, for all h, k ∈ {1, 2, … , n} and for all t ≥ 0, then the nonconservative system ( 4) is reduced to the conservative one (2).
) is a suitable initial data, the Cauchy problem related to the nonconservative framework (4) writes Remark 2. The main novelty of the nonconservative model ( 3) developed in the current paper is that the nonconservative rates are time-dependent functions.This feature renders the application of the model less restrictive and assumes relevant interest in many situations, in particular when the model is used to describe processes during long periods of time.

ANALYTICAL RESULTS
This section aims at proving some analytical results towards the Cauchy problem ( 5) and its properties.The results here presented depend on the particular functional form of the nonconservative rates  hk (t), for h, k ∈ {1, 2, … , n}.According to the previous structures, the next theorem follows.
Therefore, there exists a unique local solution f(t) ∈ (C ([0, T))) n , for T > 0, which is positive. where Specifically, this operator is Lipschitz.Let f, g ∑ n i=1 g i ≤ , for such a positive constant  > 0. Indeed, for i ∈ {1, 2, … , n}, one has Then, (7) where the constant C > 0 depends, among the others, on the initial data Then, there exists a unique local solution for T > 0 of the Cauchy problem (5), that is, For the positivity of solution f(t) (i.e.,  i (t) ≥ 0, for all i ∈ {1, 2, … , n} and for all t > 0) the formulation (5) has to be considered.Specifically, two following operators are taken into account: Bearing these expressions in mind, (4) rewrites Let us now introduce Equation ( 9) reads The operator S i [f](t) is positive, for all i ∈ {1, 2, … , n} and for all t > 0, since  hk ≥ 0, for all h, k ∈ {1, 2, … , n}, each B i hk is a probability, for all i, h, k ∈ {1, 2, … , n}, and the initial data is positive, that is,  0 i ≥ 0, for all i ∈ {1, 2, … , n}.Then, the positivity of exponential function and again the fact that initial data f 0 is positive ensure that the solution  i (t) is positive, for all i ∈ {1, 2, … , n} and for all t ≥ 0, by using expression (10).This concludes the proof.□ Theorem 1 states a rather general result on the local existence of the solution to the Cauchy problem (5), provided that the nonconservative rates are bounded.In general, Theorem 1 does not ensure a global solution of the Cauchy problem (5) since blow-up phenomena may appear due to proliferative/destructive interactions.It is worth pointing out that Theorem 1 provides a local result under very general assumptions.Although global results in time could have been obtained, they would have required too restrictive assumptions that are not realistic for the modeling objectives of kinetic theory.
Nevertheless, a local result is adequate to support the numerical simulations that will be performed in Section 4. Indeed, we consider different scenarios in which the local solution could be extended globally in time.
If the local solution f(t) is bounded for all t > 0, then the global existence is gained.In order to study the boundedness in time of solution of the Cauchy problem (5), we pass to the integral formulation of (3).Specifically, for i ∈ {1, 2, … , n}, Summing on i ∈ {1, 2, … , n} (11), one has The initial data f 0 is a vector of ( R + ) n , then the quantity ∑ n i=1  0 i is bounded.Moreover, the assumption n}, ensures that the second term on the right-hand side of ( 12) is zero.Finally, (12) The boundedness of solution is assured if the following term is bounded, for all t > 0, However, it is enough to prove the boundedness of for all i, k ∈ {1, 2, … , n}, and for all t > 0.
In general, there is not an immediate characterization of analytical request (14), since it depends on the form of coefficients  ik (t), for i, k ∈ {1, 2, … , n}.However, in this paper, three suitable scenarios are considered for this condition such that a global result is or could be gained.
First, the conservative case satisfies the boundedness condition in (14).Indeed, if for all t > 0. Therefore, if the proliferative/destructive terms  ik (t) satisfy condition (16), for all t > 0, then the system ( 5) is conservative, and there exists a unique positive and bounded solution, that is, the solution exists globally in time.In particular, if the initial data then, for all t > 0, Therefore, the solution can be regarded as a probability.However, time-dependent parameters  ik (t), for i, k ∈ {1, 2, … , n}, determine generally a nonconservative evolution of the system, that is, In what follows, we discuss two possible situations that could allow to keep bounded the term ( 14) through some assumptions on nonconservative rates  hk (t), for h, k ∈ {1, 2, … , n}, such that a global in time solution of kinetic framework (3) could be ensured.These assumptions are not restrictive for applications.Moreover, these situations provide some scenarios numerically investigated in the next section.
A first nonconservative scenario for the possible boundedness of the term ( 14) considered in this paper corresponds to a time-decay shape of the coefficients  hk (t).In particular, if for all h, k ∈ {1, 2, … , n}, with a suitable velocity, ( 14) could be bounded, and there could exist a unique global solution of the Cauchy problem (5).Even if we do not provide analytical results in this paper, this scenario is interesting for some applications.For instance, in a socio-economic system the discrete activity variable has the meaning of wealth, then the functional subsystems represent the wealth classes of society (poor, working class, middle class, rich, and so on).A realistic description needs nonconservative interactions due to the fact that, among the others, the interactions among few rich people may increase the number of poor people and reduce that of middle class.Nevertheless, these interactions are not long-lasting in time.For instance, there are some shocking events (like a crisis) that appear and allow these nonconservative interactions to become evident.However, these events are limited in time, and their influence is time decay.
Another nonconservative scenario considered in this paper for the possible boundedness of the term ( 14) corresponds to periodic nonconservative rates  hk (t).In particular, some assumptions towards functions  hk (t) can be assumed.For instance, (i) they are periodic, that is, there exists T > 0 such that  hk (s + T) =  hk (s), ∀s > 0; (ii) they have zero average, that is, (iii) they have an alternated sign, that is, for h, k ∈ {1, 2, … , n},  hk (t) ≥ 0, for t ∈ and In general, the period T depends on the particular nonconservative rate  hk (t), for h, k ∈ {1, 2, … , n}.We do not provide analytical results in this case either.From an application viewpoint, let consider the evolution of some populations in a certain environment.In general, only conservative interactions among individuals of these populations are considered for the evolution of the entire system.Moreover, nonconservative interactions, as well as birth/death processes, have a constant rate.Nevertheless, a realistic description of the dynamics needs nonconstant rates, due to the fact that some processes evolve in time and do not occur with constant rates.For instance, the impact of environment has a periodic shape related, among others, to rainfall levels, temperatures, and soil fertility, and the interaction rates could not be constant for these applications.
It is worth stressing that the above scenarios do not ensure the global existence of a solution of the kinetic framework (3); indeed, analytical results are not here presented.However, they show possible situations for the boundedness of term (14), that in turn can ensure global in time existence of a solution.In the next section, we numerically investigate these scenarios.

NUMERICAL SIMULATIONS
This section aims at presenting an application of the new kinetic framework (5) with time-dependent proliferative/destructive rates.Some numerical simulations are performed by using MATLAB routines.
According to the aims already stated in Section 1, the following numerical simulations are not related to a specific application of the new nonconservative kinetic framework (3).The reason is twofold.On one hand, the generality and versatility of the model are preserved, and its mathematical consistency is strengthen.On the other, we can discuss various scenarios, where different analytical shapes of solutions emerge, along with different long-time behaviors.Moreover, peculiar patterns appear with respect to the different choice of nonconservative parameters; therefore, the solution f(t) is determined in its behavior shape by the analytical form of these parameters.We require only some basic regularity properties such that the assumptions of Theorem 1 are ensured, that is, the local in time existence of a unique and positive solution.
First, a generic stochastically interacting system is considered.In particular, it is composed of two functional subsystems that we call populations.Then,  1 (t) and  2 (t) are the distribution functions of the first and second population, respectively.
For the sake of simplicity, the interaction rates are uniform, that is, The transition probabilities B i hk , for i, h, k ∈ {1, 2}, are chosen according to the Tables 1 and 2 such that the assumption (1) is satisfied.It is worth stressing that these choices are not restrictive, since this paper mainly focuses on the nonconservative parameters and the related dependence of solution.
As initial data In what follows, four different scenarios are presented and discussed according to the specific choice of analytical shapes for the time-dependent nonconservative rates  hk (t).

Conservative scenario
In this first scenario, the rates are such that
As well known, in the conservative case, the solution exists globally in time.The solutions  1 (t) and  2 (t) converge to an equilibrium g = (g 1 , g 2 ) after a transient time.It is worth pointing out that, in general, an explicit expression of the stationary solution depending on the parameters of the system is not available, even in the conservative case.Nevertheless, in this example, it can be computed explicitly by solving a quadratic algebraic system, due to the particular choice of the parameters and the fact that only two functional subsystems are taken into account.The equilibrium writes, explicitly and numerically, as ) , g ≈ (0.4384, 0.5616) .

Nonconservative scenario: Exponentially decay proliferative/destructive rates
Let now consider a scenario with nontrivial proliferative/destructive parameters, that is, they are not all zero.In a first case, these parameters have an exponentially decay shape, that is, In particular, the time-dependent nonconservative parameters  hk (t) are positive, for all t ≥ 0, due to the positivity of the exponential function.Therefore, in this first case, the nonconservative terms consist of proliferative rates only.The related Cauchy problem (5) satisfies Theorem 1, then there exists a unique positive solution f(t) = ( 1 (t),  2 (t)), at least bounded locally in time.The distributions  1 (t) and  2 (t) are shown in Figure 2.
Since for some values the distribution  2 (t) > 1, the solution f(t) cannot be still regarded as a probability.This is a consequence of the nonconservative structure of the framework.
Compared to the conservative scenario, now the second population shows a slight peak, after that the convergence to the equilibrium g appears.Specifically, g = (0.6021, 0.7712).

FIGURE 2
The solution f(t) = ( 1 (t),  2 (t)) of the nonconservative scenario, with exponentially decay proliferative/destructive rates given in (18).[Colour figure can be viewed at wileyonlinelibrary.com]FIGURE 3 The solution f(t) = ( 1 (t),  2 (t)) of the nonconservative scenario, with uniform periodic proliferative/destructive parameters given in (19).[Colour figure can be viewed at wileyonlinelibrary.com]Then, the impact of proliferative/destructive rates changes the solution only in a first part of the evolution of the system.Indeed, after a transient time, due to the fast decay of the exponential function, the parameters  hk (t) ≈ 0, then the behavior of solution is very close to the previous conservative framework.

Nonconservative scenario: Uniform periodic proliferative/destructive rates
In some applications, nonconservative parameters acquire a periodic structure.Therefore, in this second scenario, we consider  hk (t) with the following shape: The rates depend on time t, but they all have the same shape.In particular, they are periodic, with same period T = 1.
The related Cauchy problem (5) satisfies Theorem 1, then there exists a unique positive solution f(t) = ( 1 (t),  2 (t)), at least bounded locally in time.The distributions  1 (t) and  2 (t) are shown in Figure 3.
The two distributions  1 (t) and  2 (t) show oscillations during the evolutionary dynamics.Moreover, there is a sort of periodic nonequilibrium stationary solution, typical of kinetic scheme under the action of an external force field that plays the role of source or sink.
If the proliferative/destructive rates are the solution f(t) = ( 1 (t),  2 (t)) has a bigger amplitude, as shown in Figure 4.

Nonconservative scenario: Nonuniform periodic proliferative/destructive rates
Now, we consider another scenario with different periodic proliferative/destructive parameters.In particular, they are chosen according to Table 3.Even if the parameters  hk (t) are all related to the sinus function, they all have different period.
The related Cauchy problem (5) satisfies Theorem 1, then there exists a unique positive solution f(t) = ( 1 (t),  2 (t)), at least bounded locally in time.The distributions  1 (t) and  2 (t) are shown in Figure 5.
The distribution functions  1 (t) and  2 (t) of the two populations show a different oscillatory pattern with respect to the previous periodic case.Moreover, the solutions seem to converge to a periodic stationary state.In particular, with respect to the previous uniform periodic cases, some smaller oscillations appear during the overall evolution.

Nonconservative scenario: Oscillatory time decaying proliferative/destructive rates
Finally, we consider a situation where the proliferative/destructive rates have an oscillatory-decaying shape.In particular, The rates (21) show a periodic structure due to the presence of uniform periodic coefficient sin (4t).However, this structure decays along the time, due to the exponential term exp (2 − 0.15t).
The related Cauchy problem (5) satisfies Theorem 1, then there exists a unique positive solution f(t) = ( 1 (t),  2 (t)), at least bounded locally in time.The distributions  1 (t) and  2 (t) are shown in Figure 6.Therefore, for t → +∞, the solution f(t) converges to an equilibrium g.This situation differs from the previous periodic cases since the long-time periodic shape is not preserved during the evolution of the system, and the solution converges to a stationary state.

CONCLUSIONS AND RESEARCH PERSPECTIVES
A new nonconservative kinetic framework, (4), has been presented and discussed, both analytically and numerically.The novelty of this framework is related to the presence of nonconservative parameters  hk (t), for h, k ∈ {1, 2, … , n}, that rule time-dependent proliferative/destructive events.This choice represents the main novelty of this paper, with related analytical results and numerical simulations.
In model (3), while we lack stochastic binary nonconservative interactions, which are of interest, these interactions are, however, time dependent.As far as we know, this represents the first attempt in this direction for these specific models.Additionally, its overall generality is also a relevant aspect.
Theorem 1 provides existence and uniqueness of a positive and bounded function f(t) ∈ (C ([0, T))) n , for T > 0, which is solution of the Cauchy problem (5), once a positive suitable initial data f 0 ∈ (R + ) n is assigned.Therefore, this solution is defined only locally in time, in a time interval [0, T] depending on both the initial data f 0 and parameters of the system.Indeed, globally in time existence is not ensured since blow-up phenomena may occur.In particular, the solution could not remain bounded during the evolution of the system.Nevertheless, further analytical shapes could ensure the boundedness of solution, and the global existence of the solution can be gained.For instance, exponentially decay and periodic nonconservative parameters may help for this aim.
The new kinetic framework (4) has been tested numerically.Nevertheless, in order to not lose the generality, we have considered a generic stochastically interacting systems, composed of two populations Therefore, the overall system is divided into two functional subsystems, described by suitable distribution functions  1 (t) and  2 (t), respectively.The values of the conservative parameters of the system, that is, interaction rates and transition probabilities, are assigned, and they are fixed due to the aims of the paper.Moreover, the initial data f 0 is chosen.Then, five different scenarios are analyzed, with five different shapes of the nonconservative parameters  hk (t), for h, k ∈ {1, 2, … , n}, according to the considerations of Section 3 that may ensure global boundedness of solution.Then, the shape and behavior of the solution f(t) = ( 1 (t),  2 (t)), for each scenario, are numerically presented.It is worth stressing that these simulations do not mimic any particular applied situation, and the reason is twofold.On one hand, we point out the generality of the framework such that it could be applied in real scenarios; indeed, the new time-dependent interaction rates  hk (t) do not have a fixed analytical shape.On the other, we aim at presenting the behavior of solution with respect to time-dependent nonconservative parameters, which are the main novelty from a kinetic modeling viewpoint.In particular, this choice further point out the dependence of the shape and long-time behavior of solution with respect to these latter parameters.In Section 4.1, the conservative case is performed, that is,  hk (t) = 0, for all h, k ∈ {1, 2, … , n} and for all t ≥ 0. The solution converges to an equilibrium state.In the exponentially decay case (Section 4.2), the solution still converges to an equilibrium, even if the shape is slightly different with respect to the conservative case in the first steps of the evolution.The further two scenarios discussed in Sections 4.3 and 4.4 are characterized by the choice of periodic nonconservative parameters.In the first case, they are uniform; in the latter, they are not.For both cases, oscillatory patterns appear during the evolution.Roughly speaking, the behavior is not so different with respect to the previous cases, but a periodic structure seems to appear after some time steps.Finally, in Section 4.5, a mixed situation is presented.In particular, the coefficients  hk (t) have an oscillatory-decaying shape.Some oscillations appear, but after some iterations, the solution f(t) converges to an equilibrium state.
Even if these simulations, with related solutions, do not emerge from a specific situation, a quick interpretation can be taken, among others, from ecology.In that case, the overall system can represent an environment with two animal populations, each represented by the related distribution function  i (t), for i ∈ {1, 2}.The conservative parameters, that is, transition probability and interaction rate, describe conservative interactions between pairs of individuals, whereas the time-dependent nonconservative parameters  hk (t), for h, k ∈ {1, 2}, model the effects of proliferative/destructive events related to the natural lifetime of this population.Roughly speaking, they may be interpreted as the consequence of the existence of these populations in an open system subjected, among others, to climate change and limited food resources.For instance, the exponentially decay nonconservative rate (18) may represent an event that act on the evolution of populations, whose duration is limited in time.Indeed, after some first iterations, the solution f(t) = ( 1 (t),  2 (t)) converges to equilibrium, whereas the periodic nonconservative rates (19) and (20) and the one provided in Table 3 furnish nonconservative periodic events on populations.They may represent periodic birth/death events due to several causes, for instance, to overpopulation phenomena.Indeed, the solution shows permanent oscillations.Finally, the mixed case (21) mimic periodic events that act on the evolution of the two populations, but with a time decay structure.For instance, it may represent a periodic phenomenon on the populations that vanish after a certain amount of time.Indeed, the long-time behavior of solution shows that oscillations disappear and the equilibrium is reached.It is worth noting that this is just a first possible interpretation.
The research of analytical conditions that ensure the boundedness of solution of the Cauchy problem (5), and its consequent global existence, represents an important research perspective.Nevertheless, we aim at deriving global results under assumptions that are not too restrictive, since we do not want to undermine the realistic description of the interacting system under investigation.It is worth stressing that the system of nonlinear ordinary equations (3) is nonautonomous; therefore, some further technical difficulties emerge.Additionally, it could be of interest the study of a model with a continuous microscopic variable, that is, the activity u acquires its values in a continuous subset of R.Then, the system (3) becomes a system of nonlinear integro-differential equations, with quadratic nonlinearities.Nevertheless, we believe that some further analytical and numerical difficulties arise in that case.Then, it is of interest to understand how fast the parameters must decay in order to get a global in time existence result, starting from numerical simulations developed in that paper.Moreover, the application of the nonconservative framework (3) to real situations, by using real data, is another relevant research perspective.According to specific situation, it could be suitable the use of a different shape for the time-dependent nonconservative parameters.For instance, they could depend on the current state of each functional subsystem, through the vector distribution function f(t) = ( 1 (t),  2 (t), … ,  n (t)).Nevertheless, this might require some different analytical considerations with respect to the ones developed in this paper.Furthermore, the periodic patterns suggest possible bifurcations in this model, and this may represent material for a future work.

TABLE 3
Values of the proliferative/destructive rates  hk for nonuniform periodic case.