A Class of Solutions for the Hybrid Kinetic Model in the Tumor-Immune System Competition

Abstract

In this paper, the hybrid kinetic models of tumor-immune system competition are studied under the assumption of pure competition. The solution of the coupled hybrid system depends on the symmetry of the state transition density which characterizes the probability of successful occurrences. Thus by defining a proper transition density function, the solutions of the hybrid system are explicitly computed and applied to a classical (realistic) model of competing populations.

1. Introduction

In this paper, the two-scale tumor immune-system competition hybrid model [1–6] is studied under the assumption that the transition density function is a symmetric and separable function. The competition between tumor and immune-system can be modeled at different scales. Cells of different populations are characterized by biological functions heterogeneously distributed, and they are represented by some probability distributions. The interacting system is characterized at a macroscopic scale by a density distribution function which describes the cells activity during the interaction proliferation. At this level, the distribution of cells fulfills some partial differential equations taken from the classical kinetic theory. In this case, the more general model consists in a nonlinear system of partial differential equations. From the solution of this system, one can define a parameter which defines the time evolving distance between the two distributions, and this parameter is the charactering coefficient of the microscopic equations, typically an ordinary differential system for the competition of two populations.

This parameter has been considered [4, 5] as a random coefficient whose probability density distribution is modeled by the hiding-learning dynamics referred to biological events where tumor cells attempt to escape from immune cells which, conversely, attempt to learn about their presence.

Therefore, when the coupling parameter is obtained by solving the kinetic equations for the distribution functions, then it will be included in the classical Lotka-Volterra competition equations. We will analyze on a concrete example the influence of this stochastic parameter on the evolution. This method can be easily extended to more realistic competition models (see, e.g., [7–20]).

2. The Hybrid Model for the Tumor-Immune System Competition

Let us consider a physical system of two interacting populations, each one constituted by a large number of active particles with sizes: 𝑛_𝑖=𝑛_𝑖(𝑡),𝑛_𝑖(𝑡)∶[0,𝑇]⟶ℝ₊ (1) for 𝑖=1,2 and ℝ=₊^def[0,+∞).

Particles are homogeneously distributed in space, while each population is characterized by a microscopic state, called activity, denoted by the variable 𝑢. The physical meaning of the microscopic state may differ for each population. We assume that the competition model depends on the activity through a function of the overall distribution: 𝜇=𝜇𝑓_𝑖(𝑡,𝑢),𝜇𝑓_𝑖(𝑡,𝑢)∶ℝ₊⟶ℝ₊. (2)

The description of the overall distribution over the microscopic state within each population is given by the probability density function: 𝑓_𝑖=𝑓_𝑖(𝑡,𝑢),𝑓_𝑖(𝑡,𝑢)∶[0,𝑇]×𝐷_𝑢⟶ℝ₊,𝐷_𝑢⊆ℝ (3) for 𝑖=1,2, such that 𝑓_𝑖(𝑡,𝑢)𝑑𝑢 denotes the probability that the activity 𝑢 of particles of the 𝑖th population, at the time 𝑡, is in the interval [𝑢,𝑢+𝑑𝑢]: 𝑑𝜇=𝑓_𝑖(𝑡,𝑢)𝑑𝑢. (4) Moreover, it is ∀𝑖,∀𝑡≥0∶0≤𝑓_𝑖(𝑡,𝑢)≤1,_𝐷𝑢𝑓_𝑖(𝑡,𝑢)𝑑𝑢=1. (5)

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 refers to a specific case where the second population attempts to learn about the first population which escapes by modifying its appearance. The hybrid evolution equations specifically can be formally written as follows [4, 5]: 𝑑𝑛_𝑖𝑑𝑡=𝐺_𝑖𝑛₁,𝑛₂;𝜇𝑓,𝜕𝑓_𝑖𝜕𝑡=𝒜_𝑖𝑓, (6) where 𝐺_𝑖, for 𝑖=1,2, is a function of 𝑛={𝑛₁,𝑛₂} and 𝜇 acts over 𝑓={𝑓₁,𝑓₂}, while 𝒜_𝑖, for 𝑖=1,2, is a nonlinear operator acting on 𝑓 and 𝜇[𝑓] is a functional (0≤𝜇≤1) which describes the ability of the second population to identify the first one. Then, (6) denotes a hybrid system of a deterministic system coupled with a microscopic system statistically described by a kinetic theory approach. In the following the evolution of density distribution will be taken within the kinetic theory.

The derivation of (6)₂ can be obtained starting from a detailed analysis of microscopic interactions. Consider binary interactions specifically between a test, or candidate, particle with state 𝑢_∗ belonging to the 𝑖th population and field particle with state 𝑢^∗ belonging to the 𝑗th population. The modelling of microscopic interactions is supposed to lead to the following quantities.(i)The encounter rate, which depends for each pair of interacting populations on a suitable average of the relative velocity 𝜂_𝑖𝑗, with 𝑖, 𝑗=1,2.(ii)The transition density function 𝜑_𝑖𝑗(𝑢_∗,𝑢^∗,𝑢), which is such that 𝜑_𝑖𝑗(⋅;𝑢) denotes the probability density that a candidate particle with activity 𝑢_∗ belonging to the 𝑖th population falls into the state 𝑢∈𝐷_𝑢, of the test particle, after an interaction with a field entity, belonging to the 𝑗th population, with state 𝑢^∗. The transition density 𝜑_𝑖𝑗(𝑢_∗,𝑢^∗,𝑢) fulfills the condition ∀𝑖,𝑗,∀𝑢_∗,𝑢^∗∶_𝐷𝑢𝜑_𝑖𝑗𝑢_∗,𝑢^∗𝜑,𝑢𝑑𝑢=1,_𝑖𝑗𝑢_∗,𝑢^∗,𝑢>0, (7)when 𝜑_𝑖𝑗(𝑢_∗,𝑢^∗,𝑢)≠0 and ∀𝑢_∗,𝑢^∗∶_𝐷𝑢𝜑_𝑖𝑗𝑢_∗,𝑢^∗,𝑢𝑑𝑢=0⟺𝜑_𝑖𝑗𝑢_∗,𝑢^∗,𝑢=0. (8)

The state transition 𝑢⟶_∗^𝑢∗𝑢 (9) follows from the mutual action of the field particle (𝐹) of the 𝑖th population on the test particle (𝑇) of the 𝑗th population and vice versa so that 𝑢_∗(𝐹)⟶^{𝑢∗(𝑇)}𝑢⟺𝑢^∗(𝑇)⟶^{𝑢∗(𝐹)}𝑢. (10) With respect to this mutual action, we can assume that this function depends on the biological model, as follows. (1) Competition within the first group and with others: particles of the 𝑖th population interact with any other particle both from its own 𝑖th population and from the 𝑗th population so that 𝜑_𝑖𝑗𝑢_∗,𝑢^∗,𝑢≠0,𝑖fixed,∀𝑗. (11) In this case, each particle of the 𝑖th population can change its state not only due to the competition with the 𝑗th population but also by interacting with particles of its own population. Instead, the individuals of the 𝑗th population change their state only due to the interaction with the other 𝑖th populations. They do not interfere with each other within their 𝑖th group. (2) Competition within the second group and with others: particles of the 𝑗th population interact with any other particles both from its own 𝑗th population and from the 𝑖th population so that 𝜑_𝑖𝑗𝑢_∗,𝑢^∗,𝑢≠0,𝑗fixed,∀𝑖. (12)(3) Full competition within a group and with others: particles of each population interact with any other particles both from its own population and from the other population so that 𝜑_𝑖𝑗𝑢_∗,𝑢^∗,𝑢≠0,∀𝑖,∀𝑗. (13)(4) Competition of two groups: particles of each population interact only with particles from the other population so that 𝜑_𝑖𝑗𝑢_∗,𝑢^∗,𝑢=0,𝑖=𝑗. (14) We can assume that this kind of competition arises when the dynamics in each population are stable and each population behaves as a unique individual.

Then, by using the mathematical approach, developed in [1, 2], it yields the following class of evolution equations: 𝜕𝑓_𝑖()=𝜕𝑡𝑡,𝑢²_{𝑗=1𝐷𝑢×𝐷𝑢}𝜂_𝑖𝑗𝜑_𝑖𝑗𝑢_∗,𝑢^∗,𝑢𝑓_𝑖𝑡,𝑢_∗×𝑓_𝑗𝑡,𝑢^∗𝑑𝑢_∗𝑑𝑢^∗−𝑓_𝑖(𝑡,𝑢)²_{𝑗=1𝐷𝑢}𝜂_𝑖𝑗𝑓_𝑗𝑡,𝑢^∗𝑑𝑢^∗,(𝑖=1,2) (15) which can be formally written as (6)₂.

3. Transition Density Function Based on Separable Functions

In this section, we give the solution of (15) under some simple assumptions on the form of the transition density (7).

3.1. On the Symmetries of the State Transition Density

We assume that the integrability condition on 𝜑_𝑖𝑗, 𝜕²𝜕𝑢_∗𝜕𝑢^∗𝜑_𝑖𝑗𝑢^∗,𝑢_∗,𝑢=0, (16) holds true. As a consequence, if we write the transition density as a linear combination of separable functions, this definition implies some symmetries which will be useful for the following computations, in particular.

Theorem 1. If one defines the transition density as 𝜑_𝑖𝑗𝑢^∗,𝑢_∗,𝑢=^def12𝑎_𝑖𝜓_𝑗𝑢_∗,𝑢+𝑎_𝑗𝜓_𝑖𝑢^∗,𝑎,𝑢_𝑖,𝑎_𝑗≥0;𝑖,𝑗=1,2 (17) with 𝜓_𝑖(𝑢^∗,𝑢),𝜓_𝑗(𝑢_∗,𝑢)>0(𝑖,𝑗=1,2), the following symmetry holds true: _𝐷𝑢𝜑_𝑖𝑗𝑢_∗,𝑢^∗,𝑢𝑑𝑢=_𝐷𝑢𝜑_𝑗𝑖𝑢^∗,𝑢_∗,𝑢𝑑𝑢. (18)

Proof. From (7), (17), we have _𝐷𝑢𝑎_𝑖𝜓_𝑗𝑢_∗,𝑢+𝑎_𝑗𝜓_𝑖𝑢^∗,𝑢𝑑𝑢=2. (19) There follows, with 𝑖=1,   𝑗=2 and 𝑖=2, 𝑗=1, _𝐷𝑢𝑎₁𝜓₂𝑢_∗,𝑢+𝑎₂𝜓₁𝑢^∗,𝑢𝑑𝑢=2,_𝐷𝑢𝑎₂𝜓₁𝑢_∗,𝑢+𝑎₁𝜓₂𝑢^∗,𝑢𝑑𝑢=2 (20) so that by a comparison of _𝐷𝑢𝑎₁𝜓₂𝑢_∗,𝑢−𝜓₂𝑢^∗,𝑢+𝑎₂𝜓₁𝑢^∗,𝑢−𝜓₁𝑢_∗,𝑢𝑑𝑢=0 (21) to be valid for all 𝑎₁,𝑎₂, that is, as a consequence of the definition (17), _𝐷𝑢𝜓₂𝑢_∗,𝑢𝑑𝑢=_𝐷𝑢𝜓₂𝑢^∗,𝑢𝑑𝑢,_𝐷𝑢𝜓₁𝑢^∗,𝑢𝑑𝑢=_𝐷𝑢𝜓₁𝑢_∗,𝑢𝑑𝑢. (22)
In particular, to fulfill (20), we can assume _𝐷𝑢𝑎₁𝜓₂𝑢_∗,𝑢𝑑𝑢=1,_𝐷𝑢𝑎₂𝜓₁𝑢^∗,𝑢𝑑𝑢=1,_𝐷𝑢𝑎₂𝜓₁𝑢_∗,𝑢𝑑𝑢=1,_𝐷𝑢𝑎₁𝜓₂𝑢^∗,𝑢𝑑𝑢=1, (23)
from which, by taking into account (22), we get _𝐷𝑢𝑎₂𝜓₁𝑢_∗,𝑢𝑑𝑢=_𝐷𝑢𝑎₂𝜓₁𝑢^∗,𝑢𝑑𝑢=1⟹_𝐷𝑢𝜓₁()1𝑤,𝑢𝑑𝑢=𝑎_2𝐷𝑢,𝑎₁𝜓₂𝑢_∗,𝑢𝑑𝑢=_𝐷𝑢𝑎₁𝜓₂𝑢^∗,𝑢𝑑𝑢=1⟹_𝐷𝑢𝜓₂()1𝑤,𝑢𝑑𝑢=𝑎₁ (24) so that, by a difference, _𝐷𝑢𝑎_𝑖𝜓_𝑗(𝑣,𝑢)−𝑎_𝑗𝜓_𝑖(𝑤,𝑢)𝑑𝑢=0,𝑣,𝑤=𝑢_∗,𝑢^∗;𝑖,𝑗=1,2. (25) Thus, according to (25), the mutual action of the state transition given by the definition (7) can be summarized by (18).

Equations (17), (18) imply that the functions 𝜓_𝑖 have to be carefully chosen so that (22), (24), and (18) are fulfilled.

In the following, we will consider a special choice for the transition density (17) as 𝜑_𝑖𝑗𝑢^∗,𝑢_∗,𝑢=^def12𝑎_𝑖𝑗𝜓𝑢_∗,𝑢+𝜓𝑢^∗,𝑎,𝑢_𝑖𝑗≥0;𝑖,𝑗=1,2 (26) so that (18) is fulfilled.

3.2. Preliminary Theorems

The special choice of 𝜑_𝑖𝑗(𝑢_∗,𝑢^∗,𝑢), as defined in (26), enables us to explicitly solve (15); however, prior to computing the analytical solutions of (15), we need to show these preliminary theorems.

Theorem 2. Let 𝑋(𝑡,𝑢) be a function satisfying _𝐷𝑢𝑋(𝑡,𝑢)𝑑𝑢=𝐾𝑓(𝑡,𝑢)<𝑀<∞;𝐾≥0 (27) and 𝜓(𝑤,𝑢) a given function for which _𝐷𝑢𝜓()𝑏𝑤,𝑢𝑑𝑢=𝑎0≤𝜓(𝑤,𝑢)≤1;𝑎≠0 (28) holds, then the equation 𝜕𝑋()𝜕𝑡𝑡,𝑢=𝑎_𝐷𝑢𝜓()𝑋()()()𝑤,𝑢𝑡,𝑤𝑑𝑤−𝑏𝑓𝑡,𝑢𝑎≥0,𝑏≥0 (29) is solved by 𝑋(𝑡,𝑢)=𝐹(𝑢)𝑒^{−(𝑏−𝑎/𝜆)𝑡}+𝐾𝐺(𝑢)(𝐾≥0), (30) where 𝐹(𝑢) is the solution of the second kind homogeneous Fredholm integral equation 𝐹(𝑢)=𝜆_𝐷𝑢𝜓(𝑤,𝑢)𝐹(𝑤)𝑑𝑤,_𝐷𝑢𝐹(𝑢)𝑑𝑢=0, (31)with  𝜆 being the eigenvalue of the integral equation, and𝐺(𝑢)=𝑎𝑏_𝐷𝑢𝜓(𝑤,𝑢)𝐺(𝑤)𝑑𝑤,_𝐷𝑢𝐺(𝑢)𝑑𝑢=1(𝑏≠0), (32) when 𝑏=0, 𝐺(𝑢) is any arbitrary function fulfilling (32)₂.

Proof. Let us first notice that in the trivial case of 𝑎=0, there is no dependence on the function 𝜓𝜕𝑋()(),𝜕𝑡𝑡,𝑢=−𝑏𝑋𝑡,𝑢)(𝑏≥0 (33) but this equation is also solved by (30) being 𝑋(𝑡,𝑢)=𝐹(𝑢)𝑒^−𝑏𝑡+𝐾𝐺(𝑢). (34) In the more general case, (31)₂, (32)₂ are direct consequence of the condition (5).
By a simple computation, (29) can be transformed into the Fredholm integral equations (31), (32).
In fact, by deriving (30), we have 𝜕𝑋𝜕𝑡⁽³⁰⁾𝑎=−𝑏−𝜆𝐹(𝑢)𝑒^{−(𝑏−𝑎/𝜆)𝑡} (35) so that (29), taking into account (30), becomes −𝑎𝑏−𝜆𝐹(𝑢)𝑒^{−(𝑏−𝑎/𝜆)𝑡}=𝑎_𝐷𝑢𝜓(𝑤,𝑢)𝐹(𝑤)𝑒^{−(𝑏−𝑎/𝜆)𝑡}(𝑤𝐹(𝑢)𝑒+𝐾𝐺)𝑑𝑤−𝑏^{−(𝑏−𝑎/𝜆)𝑡}+𝐾𝐺(𝑢), (36) that is, 𝑎𝜆𝐹(𝑢)−𝑎_𝐷𝑢𝜓(𝑤,𝑢)𝐹(𝑤)𝑑𝑤𝑒^{−(𝑏−𝑎/𝜆)𝑡}=𝐾𝑎_𝐷𝑢𝜓(𝑤,𝑢)𝐺(𝑤)𝑑𝑤−𝑏𝐺(𝑢), (37) from which (31), (32) and (30) hold true.
When 𝑏=0, from the r.h.s, we have _𝐷𝑢𝜓(𝑤,𝑢)𝑑𝑤=0,_𝐷𝑢𝐺(𝑤)𝑑𝑤=1 (38) so that 𝐺(𝑢) cannot be univocally determined.

When the initial conditions are given, we have the following corollary.

Corollary 3. Let 𝑋(𝑡,𝑢) be a function satisfying (27) that is _𝐷𝑢𝑋(𝑡,𝑢)𝑑𝑢=𝐾,𝑓(𝑡,𝑢)<𝑀<∞;𝐾≥0 (39) and 𝜓(𝑤,𝑢) a given function for which _𝐷𝑢𝜓()𝑏𝑤,𝑢𝑑𝑢=𝑎0≤𝜓(𝑤,𝑢);𝑎>0,𝑏≥0 (40) holds, then the solution of the initial value problem 𝜕𝑋()𝜕𝑡𝑡,𝑢=𝑎_𝐷𝑢𝜓()𝑋()(),𝑋(𝑤,𝑢𝑡,𝑤𝑑𝑤−𝑏𝑓𝑡,𝑢𝑡,𝑢)|_𝑡=0()()=𝑋0,𝑢𝑎≥0,𝑏≥0 (41) is as follows:
(1)𝑎>0, 𝑏>0,𝐾>0𝑋()=𝑡,𝑢𝜆𝑏/𝑎𝑋()−𝑎𝜆𝑏/𝑎−10,𝑢𝑏_𝐷𝑢𝜓(𝑤,𝑢)𝑓(0,𝑤)𝑑𝑤𝑒^{−(𝑏−𝑎/𝜆)𝑡}+1𝑋()1−𝜆𝑏/𝑎0,𝑢−𝜆_𝐷𝑢𝜓()𝑋(),𝑋()𝑤,𝑢0,𝑤𝑑𝑤𝜆≠1,𝑡,𝑢=0,𝜆=1, (42)(43)
(2)𝑎>0, 𝑏>0, 𝐾=0.
The solution 𝑋(𝑡,𝑢)=𝑋(0,𝑢)𝑒^{−(𝑏−𝑎/𝜆)𝑡} (44) exists only for 𝑋(0,𝑢)=𝜆∫_𝐷𝑢𝜓(𝑤,𝑢)𝑋(0,𝑤)𝑑𝑤,
(3)𝑎>0, 𝑏≥0. The solution 𝑋(𝑡,𝑢)=𝑋(0,𝑢)𝑒^{𝑎/𝜆𝑡} (45) exists only for 𝑋(0,𝑢)=𝜆∫_𝐷𝑢𝜓(𝑤,𝑢)𝑋(0,𝑤)𝑑𝑤,
(4)𝑎=0, 𝑏≥0. For 𝐾>0, the solution of (41) does not exist. When 𝐾=0, the solution is 𝑋(𝑡,𝑢)=𝑋(0,𝑢)𝑒^−𝑏𝑡. (46)

Proof. According to Theorem 2, the solution of (41)₁ is (30) with derivative (35). In the more general case, these two equations, at the initial time, give 𝑋()(𝑢)(𝑢),−𝑎0,𝑢=𝐹+𝐾𝐺𝑏−𝜆𝐹(𝑢)=𝑎_𝐷𝑢𝜓(𝑤,𝑢)𝑋(0,𝑤)𝑑𝑤−𝑏𝑋(0,𝑢) (47) having taken into account (41)₁.
The proof of all cases above is followed by solving these two equations in 𝐹(𝑢),𝐺(𝑢) with respect to the initial condition 𝑋(0,𝑢).
For instance, for the first case (1), there follows (𝑢)()(𝑢),𝐹(𝑢)1𝐾𝐺=𝑋0,𝑢−𝐹=−𝑏/𝑎−1/𝜆_𝐷𝑢𝜓()𝑋()+𝑤,𝑢0,𝑤𝑑𝑤𝑏/𝑎𝑋(),𝑏/𝑎−1/𝜆0,𝑢 (48) that is (𝑢)=1𝐾𝐺𝑋()1−𝜆𝑏/𝑎0,𝑢−𝜆_𝐷𝑢𝜓()𝑋(),𝐹(𝑢)=𝑤,𝑢0,𝑤𝑑𝑤𝜆𝑏/𝑎𝑋()−𝑎𝜆𝑏/𝑎−10,𝑢𝑏_𝐷𝑢𝜓(𝑤,𝑢)𝑋(0,𝑤)𝑑𝑤 (49) so that (42) holds true.
When 𝑋()=𝑎0,𝑢𝑏_𝐷𝑢𝜓(𝑤,𝑢)𝑋(0,𝑤)𝑑𝑤 (50) which implies 𝜆=1, from (49), we get a trivial solution of (29), (31), (32) and (47); that is, 𝐹(𝑢)=0,𝐺(𝑢)=0. (51)
Analogously, for the case (2) system (47) becomes 𝑋()(𝑢)(𝑢),(𝑢)0,𝑢=𝐹+𝐾𝐺−𝐹=𝜆_𝐷𝑢𝜓(𝑤,𝑢)𝑋(0,𝑤)𝑑𝑤. (52) However, if 𝐾≠0, the integral of the right side of the second equation is 𝐾, while the integral of the first side must be zero.
With similar reasonings, we get the proof of the remaining cases.

4. Solution of the System (15)

In this section, we will give the explicit solution of the system (15) under some suitable hypotheses on both the encounter rate 𝜂_𝑖𝑗 and the transition density 𝜑_𝑖𝑗(𝑢_∗,𝑢^∗,𝑢). Let us assume the symmetry of 𝜂_𝑖𝑗 so that 𝜂=₁^def𝜂₁₁,𝜂=₂^def𝜂₂₂,𝜂=₀^def𝜂₁₂=𝜂₂₁. (53)

Thanks to the previous theorems, and the symmetry of 𝜑_𝑖𝑗(𝑢_∗,𝑢^∗,𝑢) as given by (18), system (15) simplifies, the following.

Theorem 4. Let the transition density 𝜑_𝑖𝑗(𝑢_∗,𝑢^∗,𝑢) be defined as 𝜑_𝑖𝑗𝑢_∗,𝑢^∗=1,𝑢2𝑎_𝑖𝑗𝜓𝑢_∗,𝑢+𝜓𝑢^∗,𝑎,𝑢𝑖,𝑗=1,2_1def=𝑎₁₁,𝑎=₂^def𝑎₂₂,𝑎=₀^def𝑎₁₂=𝑎₂₁, (54) which fulfills (7) and the symmetries conditions (18), and the density function 𝜓(𝑤,𝑢) such that 12𝑎_{𝑖𝑗𝐷𝑢}𝜓𝑢_∗,𝑢+𝜓𝑢^∗,𝑢𝑑𝑢=⁽⁷⁾1 (55) holds. Equation (15) can be simplified into 𝜕𝑓₁()𝜕𝑡𝑡,𝑢=𝜂₁𝑎_1𝐷𝑢𝜓(𝑤,𝑢)𝑓₁()+1𝑡,𝑤𝑑𝑤2𝜂₀𝑎_0𝐷𝑢𝜓(𝑤,𝑢)𝑓₁(𝑡,𝑤)+𝑓₂(𝑡,𝑤)𝑑𝑤−𝑓₁(𝑡,𝑢)𝜂₁+𝜂₀,𝜕𝑓₂()=1𝜕𝑡𝑡,𝑢2𝜂₀𝑎_0𝐷𝑢𝜓(𝑤,𝑢)𝑓₂(𝑡,𝑤)+𝑓₁(𝑡,𝑤)𝑑𝑤+𝜂₂𝑎_2𝐷𝑢𝜓(𝑤,𝑢)𝑓₂(𝑡,𝑤)𝑑𝑤−𝑓₂(𝑡,𝑢)𝜂₀+𝜂₂. (56)

Proof. By a substitution of (54) into (15), we get 𝜕𝑓₁()=𝜕𝑡𝑡,𝑢²_{𝑗=1𝐷𝑢×𝐷𝑢}𝜂_1𝑗𝑎_1𝑗𝜓𝑢_∗,𝑢+𝜓𝑢^∗,𝑢×𝑓₁𝑡,𝑢_∗𝑓_𝑗𝑡,𝑢^∗𝑑𝑢_∗𝑑𝑢^∗−𝑓₁(𝑡,𝑢)²_{𝑗=1𝐷𝑢}𝜂_1𝑗𝑓_𝑗𝑡,𝑢^∗𝑑𝑢^∗,𝜕𝑓₂()=𝜕𝑡𝑡,𝑢²_{𝑗=1𝐷𝑢×𝐷𝑢}𝜂_2𝑗𝑎_2𝑗𝜓𝑢_∗,𝑢+𝜓𝑢^∗,𝑢×𝑓₂𝑡,𝑢_∗𝑓_𝑗𝑡,𝑢^∗𝑑𝑢_∗𝑑𝑢^∗−𝑓₂(𝑡,𝑢)²_{𝑗=1𝐷𝑢}𝜂_2𝑗𝑓_𝑗𝑡,𝑢^∗𝑑𝑢^∗, (57) that is, 𝜕𝑓₁()=1𝜕𝑡𝑡,𝑢2_{𝐷𝑢×𝐷𝑢}𝜂₁₁𝑎₁₁𝜓𝑢_∗,𝑢+𝜓𝑢^∗,𝑢×𝑓₁𝑡,𝑢_∗𝑓₁𝑡,𝑢^∗𝑑𝑢_∗𝑑𝑢^∗+_{𝐷𝑢×𝐷𝑢}𝜂₁₂𝑎₁₂𝜓𝑢_∗,𝑢+𝜓𝑢^∗,𝑢×𝑓₁𝑡,𝑢_∗𝑓₂𝑡,𝑢^∗𝑑𝑢_∗𝑑𝑢^∗−𝑓₁(𝑡,𝑢)_𝐷𝑢𝜂₁₁𝑓₁𝑡,𝑢^∗𝑑𝑢^∗+_𝐷𝑢𝜂₁₂𝑓₂𝑡,𝑢^∗𝑑𝑢^∗,𝜕𝑓₂()=1𝜕𝑡𝑡,𝑢2_{𝐷𝑢×𝐷𝑢}𝜂₂₁𝑎₂₁𝜓𝑢_∗,𝑢+𝜓𝑢^∗,𝑢×𝑓₂𝑡,𝑢_∗𝑓₁𝑡,𝑢^∗𝑑𝑢_∗𝑑𝑢^∗+_{𝐷𝑢×𝐷𝑢}𝜂₂₂𝑎₂₂𝜓𝑢_∗,𝑢+𝜓𝑢^∗,𝑢×𝑓₂𝑡,𝑢_∗𝑓₂𝑡,𝑢^∗𝑑𝑢_∗𝑑𝑢^∗−𝑓₂(𝑡,𝑢)_𝐷𝑢𝜂₂₁𝑓₁𝑡,𝑢^∗𝑑𝑢^∗+_𝐷𝑢𝜂₂₂𝑓₂𝑡,𝑢^∗𝑑𝑢^∗, (58) from which, 𝜕𝑓₁()=1𝜕𝑡𝑡,𝑢2𝜂₁₁𝑎_{11𝐷𝑢×𝐷𝑢}𝜓𝑢_∗,𝑢𝑓₁𝑡,𝑢_∗×𝑓₁𝑡,𝑢^∗𝑑𝑢_∗𝑑𝑢^∗+_{𝐷𝑢×𝐷𝑢}𝜓𝑢^∗,𝑢𝑓₁𝑡,𝑢_∗×𝑓₁𝑡,𝑢^∗𝑑𝑢_∗𝑑𝑢^∗+12𝜂₁₂𝑎_{12𝐷𝑢×𝐷𝑢}𝜓𝑢_∗,𝑢𝑓₁𝑡,𝑢_∗×𝑓₂𝑡,𝑢^∗𝑑𝑢_∗𝑑𝑢^∗+_{𝐷𝑢×𝐷𝑢}𝜓𝑢^∗,𝑢𝑓₁𝑡,𝑢_∗×𝑓₂𝑡,𝑢^∗𝑑𝑢_∗𝑑𝑢^∗−𝑓₁(𝑡,𝑢)𝜂_11𝐷𝑢𝑓₁𝑡,𝑢^∗𝑑𝑢^∗+𝜂_12𝐷𝑢𝑓₂𝑡,𝑢^∗𝑑𝑢^∗,𝜕𝑓₂()=1𝜕𝑡𝑡,𝑢2𝜂₂₁𝑎_{21𝐷𝑢×𝐷𝑢}𝜓𝑢_∗,𝑢𝑓₂𝑡,𝑢_∗×𝑓₁𝑡,𝑢^∗𝑑𝑢_∗𝑑𝑢^∗+_{𝐷𝑢×𝐷𝑢}𝜓𝑢^∗,𝑢𝑓₂𝑡,𝑢_∗×𝑓₁𝑡,𝑢^∗𝑑𝑢_∗𝑑𝑢^∗+12𝜂₂₂𝑎_{22𝐷𝑢×𝐷𝑢}𝜓𝑢_∗,𝑢𝑓₂𝑡,𝑢_∗×𝑓₂𝑡,𝑢^∗𝑑𝑢_∗𝑑𝑢^∗+_{𝐷𝑢×𝐷𝑢}𝜓𝑢^∗,𝑢𝑓₂𝑡,𝑢_∗×𝑓₂𝑡,𝑢^∗𝑑𝑢_∗𝑑𝑢^∗−𝑓₂(𝑡,𝑢)_𝐷𝑢𝜂₂₁𝑓₁𝑡,𝑢^∗𝑑𝑢^∗+_𝐷𝑢𝜂₂₂𝑓₂𝑡,𝑢^∗𝑑𝑢^∗. (59)
There follows 𝜕𝑓₁()=1𝜕𝑡𝑡,𝑢2𝜂₁₁𝑎_11𝐷𝑢𝜓𝑢_∗,𝑢𝑓₁𝑡,𝑢_∗𝑑𝑢_∗×_𝐷𝑢𝑓₁𝑡,𝑢^∗𝑑𝑢^∗+_𝐷𝑢𝜓𝑢^∗,𝑢𝑓₁𝑡,𝑢^∗𝑑𝑢^∗×_𝐷𝑢𝑓₁𝑡,𝑢_∗𝑑𝑢_∗+12𝜂₁₂𝑎_12𝐷𝑢𝜓𝑢_∗,𝑢𝑓₁𝑡,𝑢_∗𝑑𝑢_∗×_𝐷𝑢𝑓₂𝑡,𝑢^∗𝑑𝑢^∗+_𝐷𝑢𝜓𝑢^∗,𝑢𝑓₂𝑡,𝑢^∗𝑑𝑢^∗×_𝐷𝑢𝑓₁𝑡,𝑢_∗𝑑𝑢_∗−𝑓₁(𝑡,𝑢)𝜂_11𝐷𝑢𝑓₁𝑡,𝑢^∗𝑑𝑢^∗+𝜂_12𝐷𝑢𝑓₂𝑡,𝑢^∗𝑑𝑢^∗,𝜕𝑓₂()=1𝜕𝑡𝑡,𝑢2𝜂₂₁𝑎_21𝐷𝑢𝜓𝑢_∗,𝑢𝑓₂𝑡,𝑢_∗𝑑𝑢_∗×_𝐷𝑢𝑓₁𝑡,𝑢^∗𝑑𝑢^∗+_𝐷𝑢𝜓𝑢^∗,𝑢𝑓₁𝑡,𝑢^∗𝑑𝑢^∗×_𝐷𝑢𝑓₂𝑡,𝑢_∗𝑑𝑢_∗+12𝜂₂₂𝑎_22𝐷𝑢𝜓𝑢_∗,𝑢𝑓₂𝑡,𝑢_∗𝑑𝑢_∗×_𝐷𝑢𝑓₂𝑡,𝑢^∗𝑑𝑢^∗+_𝐷𝑢𝜓𝑢^∗,𝑢𝑓₂𝑡,𝑢^∗𝑑𝑢^∗×_𝐷𝑢𝑓₂𝑡,𝑢_∗𝑑𝑢_∗−𝑓₂(𝑡,𝑢)𝜂_21𝐷𝑢𝑓₁𝑡,𝑢^∗𝑑𝑢^∗+𝜂_22𝐷𝑢𝑓₂𝑡,𝑢^∗𝑑𝑢^∗. (60)
According to (5), we get 𝜕𝑓₁()=1𝜕𝑡𝑡,𝑢2𝜂₁₁𝑎_11𝐷𝑢𝜓𝑢_∗,𝑢𝑓₁𝑡,𝑢_∗𝑑𝑢_∗+_𝐷𝑢𝜓𝑢^∗,𝑢𝑓₁𝑡,𝑢^∗𝑑𝑢^∗+12𝜂₁₂𝑎_12𝐷𝑢𝜓𝑢_∗,𝑢𝑓₁𝑡,𝑢_∗𝑑𝑢_∗+_𝐷𝑢𝜓𝑢^∗,𝑢𝑓₂𝑡,𝑢^∗𝑑𝑢^∗−𝑓₁(𝑡,𝑢)𝜂₁₁+𝜂₁₂,𝜕𝑓₂()=1𝜕𝑡𝑡,𝑢2𝜂₂₁𝑎_21𝐷𝑢𝜓𝑢_∗,𝑢𝑓₂𝑡,𝑢_∗𝑑𝑢_∗+_𝐷𝑢𝜓𝑢^∗,𝑢𝑓₁𝑡,𝑢^∗𝑑𝑢^∗+12𝜂₂₂𝑎_22𝐷𝑢𝜓𝑢_∗,𝑢𝑓₂𝑡,𝑢_∗𝑑𝑢_∗+_𝐷𝑢𝜓𝑢^∗,𝑢𝑓₂𝑡,𝑢^∗𝑑𝑢^∗−𝑓₂(𝑡,𝑢)𝜂₂₁+𝜂₂₂. (61)
Thus, we obtain, by a variable change, 𝜕𝑓₁()𝜕𝑡𝑡,𝑢=𝜂₁₁𝑎_11𝐷𝑢𝜓(𝑤,𝑢)𝑓₁()+1𝑡,𝑤𝑑𝑤2𝜂₁₂𝑎_12𝐷𝑢𝜓(𝑤,𝑢)𝑓₁(𝑡,𝑤)+𝑓₂(𝑡,𝑤)𝑑𝑤−𝑓₁(𝑡,𝑢)𝜂₁₁+𝜂₁₂,𝜕𝑓₂()=1𝜕𝑡𝑡,𝑢2𝜂₂₁𝑎_21𝐷𝑢𝜓(𝑤,𝑢)𝑓₂(𝑡,𝑤)+𝑓₁(𝑡,𝑤)𝑑𝑤+𝜂₂₂𝑎_22𝐷𝑢𝜓(𝑤,𝑢)𝑓₂(𝑡,𝑤)𝑑𝑤−𝑓₂(𝑡,𝑢)𝜂₂₁+𝜂₂₂ (62) so that (56) follows.

4.1. Pure Competition Model

We will consider the solution of (56) when, together with the hypotheses (53), (54)₂, some more conditions are given on the parameters.

According to (26), let us assume 𝜑_𝑖𝑗𝑢^∗,𝑢_∗,𝑢=^def12𝑎_𝑖𝑗𝜓𝑢_∗,𝑢+𝜓𝑢^∗,𝑎,𝑢_𝑖𝑗≥0;𝑖,𝑗=1,2 (63) together with the symmetries conditions (18).

If we define 𝜂=₁^def𝜂₁₁,𝜂=₂^def𝜂₂₂,𝜂=₀^def𝜂₁₂=𝜂₂₁,𝑎=₁^def𝑎₁₁,𝑎=₂^def𝑎₂₂,𝑎=₀^def𝑎₁₂=𝑎₂₁, (64) we will discuss only the following hypotheses: 𝑎=^def𝑎₁=𝑎₂𝜂=0,₁=𝜂₂=𝜂≠0,𝜂₀𝑎₀≠0, (65)(66) which seem to have some biological interpretations, being the pure encounter-competition model. This happens when the transition of state arises only when particles of one population interact only with an individual of the other population. In this case, individuals of one population do not interact with individuals of the same population.

Theorem 5. Let the transition density 𝜑_𝑖𝑗(𝑢_∗,𝑢^∗,𝑢) be defined as 𝜑_𝑖𝑗𝑢_∗,𝑢^∗=1,𝑢2𝑎_𝑖𝑗𝜓𝑢_∗,𝑢+𝜓𝑢^∗,𝑢,𝑖,𝑗=1,2 (67) with 𝑎_𝑖𝑗 as given by (64), (65). This definition of the transition density fulfills (7) and the symmetries conditions (18). The density function 𝜓(𝑤,𝑢) is such that _𝐷𝑢𝜓()1𝑤,𝑢𝑑𝑢=𝑎. (68)
By assuming 𝑎₁=𝑎₂=0,𝜂₁=𝜂₂=𝜂≠0 (69) and for 𝑎₀,𝜂₀, the condition 𝜂₀𝑎₀≠0 (70) system (56) becomes 𝜕𝑓₁()=1𝜕𝑡𝑡,𝑢2𝜂₀𝑎_0𝐷𝑢𝜓(𝑤,𝑢)𝑓₁(𝑡,𝑤)+𝑓₂(𝑡,𝑤)𝑑𝑤−𝑓₁(𝑡,𝑢)𝜂+𝜂₀,𝜕𝑓₂()=1𝜕𝑡𝑡,𝑢2𝜂₀𝑎_0𝐷𝑢𝜓(𝑤,𝑢)𝑓₂(𝑡,𝑤)+𝑓₁(𝑡,𝑤)𝑑𝑤−𝑓₂(𝑡,𝑢)𝜂+𝜂₀, (71) and its solution is given by 𝑓₁(𝑡,𝑢)=𝑒^{−[𝜂+𝜂0]𝑡}𝐹(𝑢)𝑒^{[𝜂0𝑎0/𝜆]𝑡}(𝑢(𝑢),𝑓+𝐻)+𝐺₂(𝑡,𝑢)=𝑒^{−[𝜂+𝜂0]𝑡}𝐹(𝑢)𝑒^{[𝜂0𝑎0/𝜆]𝑡}(𝑢(𝑢),𝐹(𝑢)−𝐻)+𝐺=𝜆𝜂+𝜂₀𝜂₀𝑎_0𝐷𝑢𝜓(𝑤,𝑢)𝐹(𝑤)𝑑𝑤,_𝐷𝑢𝐹(𝑢)𝐺(𝑢)=𝜂𝑑𝑢=0,₀𝜂+𝜂₀𝑎_0𝐷𝑢𝜓(𝑤,𝑢)𝐺(𝑤)𝑑𝑤,_𝐷𝑢𝐺(𝑢)𝑑𝑢=1,_𝐷𝑢𝜓(𝑤,𝑢)𝑑𝑤=𝜂+𝜂₀𝜂₀𝑎_0𝐷𝑢,𝐻(𝑤)𝑑𝑤=0. (72)

Proof. From (56), we have 𝜕𝑓₁()=1𝜕𝑡𝑡,𝑢2𝜂₀𝑎_0𝐷𝑢𝜓(𝑤,𝑢)𝑓₁(𝑡,𝑤)+𝑓₂(𝑡,𝑤)𝑑𝑤−𝑓₁(𝑡,𝑢)𝜂+𝜂₀,𝜕𝑓₂()=1𝜕𝑡𝑡,𝑢2𝜂₀𝑎_0𝐷𝑢𝜓(𝑤,𝑢)𝑓₂(𝑡,𝑤)+𝑓₁(𝑡,𝑤)𝑑𝑤−𝑓₂(𝑡,𝑢)𝜂+𝜂₀, (73) from which by linear combination, we get 𝜕𝑓𝜕𝑡¹(𝑡,𝑢)+𝑓₂(𝑡,𝑢)=𝜂₀𝑎_0𝐷𝑢𝜓(𝑤,𝑢)𝑓₁(𝑡,𝑤)+𝑓₂(−𝑓𝑡,𝑤)𝑑𝑤₁(𝑡,𝑢)+𝑓₂(𝑡,𝑢)𝜂+𝜂₀,𝜕𝑓𝜕𝑡¹(𝑡,𝑢)−𝑓₂(𝑡,𝑢)=−𝑓₁(𝑡,𝑢)−𝑓₂(𝑡,𝑢)𝜂+𝜂₀. (74)
With the above positions, we have 𝜕𝑋()𝜕𝑡𝑡,𝑢=𝜂₀𝑎_0𝐷𝑢𝜓(𝑤,𝑢)𝑋(𝑡,𝑢)𝑑𝑤−𝑋(𝑡,𝑢)𝜂+𝜂₀,𝜕𝑌()(𝜕𝑡𝑡,𝑢=−𝑌𝑡,𝑢)𝜂+𝜂₀ (75) so that, by taking into account Theorem 2, it is 𝑋(𝑡,𝑢)=𝐹(𝑢)𝑒^{−[𝜂+𝜂0−𝜂0𝑎0/𝜆]𝑡}(𝑢),𝐹(𝑢)+𝐺=𝜆𝜂+𝜂₀𝜂₀𝑎_0D𝑢𝜓(𝑤,𝑢)𝐹(𝑤)𝑑𝑤,_𝐷𝑢𝐹(𝑢)𝐺(𝑢)=𝜂𝑑𝑢=0,₀𝜂+𝜂₀𝑎_0𝐷𝑢𝜓(𝑤,𝑢)𝐺(𝑤)𝑑𝑤,_𝐷𝑢𝐺(𝑢)𝑑𝑢=1,_𝐷𝑢𝜓(𝑤,𝑢)𝑑𝑤=𝜂+𝜂₀𝜂₀𝑎₀,𝑌(𝑡,𝑢)=𝐻(𝑢)𝑒^{−[𝜂+𝜂0]𝑡},_𝐷𝑢𝐻(𝑤)𝑑𝑤=0 (76) from which (72) follows.

Example 6. A transition density, which is compatible with this case, is the following: 𝜑_𝑖𝑗𝑢_∗,𝑢^∗=𝑎,𝑢⁰21−𝛿_𝑖𝑗𝜓𝑢_∗,𝑢+𝜓𝑢^∗,𝑢,𝑖,𝑗=1,2 (77) with 𝛿_𝑖𝑗, Kronecker symbol.

5. Application to Lotka-Volterra Model

In this section, we will study a coupled system (6) where the macroscopic equations are the Lotka-Volterra equations (6)₁. Concerning the coupling stochastic parameter 𝜇[𝑓], we have to define the functional 𝜇 in (2), (6) depending on the “distance” between distributions; that is, 𝜇𝑓_𝑖,𝑓_𝑗(𝑡)=𝜇𝑓_𝑖−𝑓_𝑗(𝑡) (78) with 0≤𝜇𝑓_𝑖,𝑓_𝑗(𝑡)≤1,∀𝑢∈𝐷_𝑢𝜇𝑓∧𝑡∈𝑇,_𝑖,𝑓_𝑗(𝑡)=1⟺𝑓_𝑖=𝑓_𝑗,𝜇𝑓_𝑖,𝑓_𝑗(𝑡)=0⟺𝑓_𝑖=0∨𝑓_𝑗=0, (79) where the maximum learning result is obtained when the second population is able to reproduce the distribution of the first one: 𝑓₁=𝑓₂, while the minimum learning is achieved when one distribution is vanishing.

In some recent papers; it has been assumed [4, 5] that 𝜇𝑓_𝑖,𝑓_𝑗(𝑡)=𝜇𝑓_𝑖−𝑓_𝑗(𝑡)=1−_𝐷𝑢𝑓₁(𝑡,𝑢)−𝑓₂(𝑡,𝑢)𝑑𝑢. (80)

In this case, it is 𝜇=1, when 𝑓₁=𝑓₂; otherwise 𝜇≠1 with 𝜇↓0, depending on the time evolution of the distance between 𝑓₁ and 𝑓₂.

Let us notice that 𝜇 is the coupling term which links the macroscopic model (6)₁ to the microscopic model (6)₂. There follows that the solution of the hybrid system (6) depends on the coupling parameter 𝜇 (80) which follows from the solution of (15). System (15) is a system of two nonlinear integrodifferential equations constrained by the conditions (7), (5). Moreover, its solution depends also the constant encounter rate 𝜂_𝑖𝑗, on the transition density function 𝜑_𝑖𝑗, and the initial conditions 𝑓_𝑖(0,𝑢). In the following section, we will study the solution of (6), under some suitable, but not restrictive, hypotheses on 𝜑_𝑖𝑗.

Under the hypotheses of Theorem 5 and the solution (72), we have 𝑓₁(𝑡,𝑢)−𝑓₂(𝑡,𝑢)=2𝑒^{−(𝜂+𝜂0)𝑡}𝐻(𝑢)(𝑢)(𝑢),𝐻≠0,0,𝐻=0. (81) Let us take 𝐷_𝑢=[0,1],𝜂+𝜂₀=𝑝>0,𝐻(𝑢)=sin2𝜋𝑢 (82) so that (72) are fulfilled. We have 𝜇𝑓(𝑡)=1−2𝑒^−𝑝𝑡₀^1/2sin2𝜋𝑢𝑑𝑢−¹_1/2sin2𝜋𝑢𝑑𝑢, (83) that is 𝜇𝑓𝑡)=1(1−𝜋𝑒^−𝑝𝑡[],1,sin2𝜋𝑢≠0,𝑢∈0,11,𝑢∈0,2,1. (84)

In the last case we have the usual Lotka-Volterra system, therefore, we will investigate the first case. Thus, according to (6), we have the system 𝑑𝑛₁𝑑𝑡=𝛼𝑛₁−11−𝜋𝑒^−𝑝𝑡𝑛₁𝑛₂,𝑑𝑛₂𝑑𝑡=−𝛽𝑛₂+𝛾𝑛₁𝑛₂ (85) with 𝛼≥0,𝛽≥0,𝛾≥0. (86) The numerical solution of this system depends on both the parameters 𝛼,𝛽,𝛾,𝑝 and on the initial conditions 𝑛₁(0), 𝑛₂(0). We can see from Figures 1 and 2 that albeit the initial aggressive population 𝑛₂ is greater than 𝑛₁, the first population can increase and keep nearly always over 𝑛₂ in the quasilinear case in Figure 1(a) or always under 𝑛₂ in presence of a strong nonlinearity in Figure 1(b).

(a)

(b)

Figure 1 

Numerical solution of 𝑛1(𝑡) (plain) and 𝑛2(𝑡) (dashed) of the system (85) with initial conditions 𝑛1(0)=1,𝑛2(0)=3, for 𝑡≤10, and parameters 𝛼=1.636,𝛽=0.3743 ((a) with parameters 𝛾=0.1, 𝑝=0.01) and ((b) with parameters 𝛾=0.9, 𝑝=0.9).

(a)

(b)

Figure 2 

Numerical solution of 𝑛1(𝑡) (plain) and 𝑛2(𝑡) (dashed) of the system (85) with initial conditions 𝑛1(0)=3,𝑛2(0)=1, for 𝑡≤10 and parameters 𝛼=1.636, 𝛽=0.3743 ((a) with parameters 𝛾=0.1, 𝑝=0.01) and ((b) with parameters 𝛾=0.9, 𝑝=0.9).

If we invert the initial conditions so that the initial population of 𝑛₁ is greater than 𝑛₂, we can see that in case of quasilinear conditions (see Figure 2(a)) the population 𝑛₁ after some short time becomes lower than 𝑛₂. For a strong nonlinearity, instead after an initial growth 𝑛₁, it tends to zero in a short time, while the second population grows very fast and becomes the prevalent population in Figure 2(b).

6. Conclusion

In this paper, the hybrid competition model has been solved under some assumptions on the transition density. In the simple case of Lotka-Volterra, the numerical solution gives some significant and realistic insights on the evolution of competing populations.