1. Introduction
Cancer is a group of diseases involving abnormal cell growth. These cells could form a mass known as a tumor. A malignant tumor means it could invade or spread into nearby tissues. Therefore, cancer cells (linked with tumor growth) could reach distant parts of the body and form a new tumor (far from the original one). A benign tumor means the tumor can grow but not spread.
Many authors have used mathematical models. The main components of these models involve interactions among three types of cells, cancer cells with healthy host cells, and cells of the immune system. These interactions may lead to different outcomes. In the literature, there are several reviews on mathematical systems applied to tumor dynamics, e.g., [
1,
2,
3,
4].
An approach using a discrete-time system is better to describe the tumor dynamics compared to a continuous-time system (when populations have non-overlapping generations). Regarding computational and numerical simulations, discrete-time models are more efficient. For the advantages of the discrete approach compared to the continuous approach, see [
4,
5,
6,
7].
Chaos can be found in many biological systems. Chaotic systems have internal behaviors that depend (in many cases) on the initial conditions and could be suppressed as a result of the application of small perturbations.
Chaotic behaviors are complex in general, full of irregularities, and are generally undesirable in biological systems. Model complexities, in many cases, need to be reduced in mathematical modeling to generate dynamic results. Discretization methods, e.g., the Taylor series expansion or Euler and Runge–Kutta methods, are prime tools used to treat chaotic systems. These methods are of particular importance when studying differential equations applied to tumor dynamics. Selecting the best discretization approach is a problem in itself. We endeavored to compare the Euler and Runge–Kutta numerical integration approach to simulate the chaotic behavior of a multi-scroll chaotic oscillator and compare the obtained results.
In this paper, we consider the model presented by Pillis and Radunskaya; see [
1]. In the second section (after the introduction), we present the cancer system in a continuous state (with some numerical results). In the third section, we present discretization methods, for example, the Euler method, the Taylor series expansion method, and the Runge–Kutta method; we applied them to the cancer system. In the fourth section, we review the stability of the fixed points in the discrete cancer system. In the fifth section, we prove that the discrete cancer system is chaotic, using the new version of Marotto’s theorem at a fixed point. In the sixth section, we present numerical simulations (e.g., Lyapunov exponents and bifurcations diagrams). Finally, we present a conclusion.
2. The Continuous Version of the Cancer System
The AIMS model is used to describe the competition and interactions among tumor cells, healthy host cells, and effector immune cells. Regarding cancer models that include interacting cells, we focused on cells near the tumor sites. Populations are based on the prey–predator models and the law of exponential growth. Although the previous models are simple, they are suitable platforms to explain important aspects of the dynamics of cancer. In this section, we will present the model to describe the biological tumor system, which is given in the form of an ordinary differential equation as follows
where
N denotes the healthy host cells,
T denotes the number of cancer cells,
I denotes the effector immune cells, and
, and
s are positive parameters; see [
1,
2,
4]. Here,
represents the growth rate (in the absence of any effect) of cancer cells from other cell populations with a maximum carrying capacity of
; the values
and
refer to the ’killing rate’ of the cancer cells by the healthy host cells and effector cells, respectively;
represents the growth rate, with a maximum carrying capacity
of healthy host cells;
represents the rate of inactivation of the healthy cells by the cancer cells. The rate (or level) of recognition of the cancer cells by the immune system depends usually on the antigenicity of the cancer cells. Due to the large complexity of this recognition process and to keep the model simpler, we assume that the stimulation of the immune system depends—in a direct way—on the number of cancer cells with positive constants,
and
. We consider that the effector cells are inactivated by the cancer cells at rate
and that they die in a natural way at rate
. The value
s is a constant influx of immune cells.
To simplify the study of this system (
1). we reduced the number of parameters by introducing this change of variables:
and the new parameters:
then system (
1) is converted to, see
Figure 1,
Figure 2,
Figure 3 and
Figure 4:
The Lyapunov exponents of system (
2) are computed to be
and
. The Lyapunov dimension for system (
2) is
4. Stability Analysis for Discrete Cancer System
To find the fixed points, the three discrete cancer equations were set to
coordinates of each fixed point, determined by solving the following equations:
To obtain the fixed points of the system (
4), we set
The solutions from Equations (
9)–(
11) together yielded to six fixed points. We discussed their local behaviors according to their biological relevance. Now, we will look at the stabilities of these fixed points.
In this paper, we studied h in interval
(1) The first fixed point is trivial and given as , the corresponding eigenvalues are , and . Since h is small positive, all the parameters are positive, and ; therefore,
Proposition 1. If then , we have a saddle at this fixed point.
If then , we have a node stable at this fixed point.
(2) The second fixed point is
; the Jacobian matrix evaluated at
is given by
Clearly, has eigenvalues , and . where . In fact, in biology, are smaller than . Then and . The stability of this fixed point depends on the value of parameter , if then , this fixed point has two stable and one unstable eigenvalue. Therefore, we have a saddle at , and if , then ; this fixed point has three stable eigenvalues. Therefore, we have a node at this fixed point. If , then ; therefore, we cannot give any information on the stability of . In the numerical simulations, we obtained different results depending on the values of . We observed that the chaotic dynamics appeared close to . The selection of provides different dynamical behaviors, such as convergence to a stable spiral. However, in this study, we focus on parameter , where we have chaotic attraction.
(3) The third fixed point is
; the Jacobian matrix evaluated at
is given by
The eigenvalues of the Jacobian matrix (
13) at this fixed point are obtained as
,
and
. Then
. Moreover,
is stable, and
,
are stable with the selected parameters.
(4) The fourth fixed point is
. The Jacobian matrix evaluated at
is given by
where
The eigenvalues of the Jacobian matrix at this point are
- (i)
If , we have three real eigenvalues.
- (ii)
If , we have one real and two complex eigenvalues that are stable with the selected parameter sets.
The characteristic equation of the Jacobian matrix
can be expressed in the form
where
According to the Jury conditions [
11], to find the asymptotically-stable region of
, it is necessary to find the region holding these conditions:
where
Then
According to the relations we have
, and .
(5) The fifth fixed point is
, where
. The Jacobian matrix of system (
4) at
is given by
where
and
The characteristic equation of the Jacobian matrix
can be written as
where
The eigenvalues of the Jacobian matrix at this fixed point are and where ,
(6) The sixth fixed point is a nontrivial
. The Jacobian matrix of system (
4) at
is given by
where
The characteristic equation of the Jacobian matrix
can be written as
According to the Jury conditions [
11], to find the asymptotically-stable region of
, we need to find the region that satisfies the following conditions:
where
Since
Proposition 2. The fixed point is asymptotically stable if the following conditions are satisfied:
, and .