Spatiotemporal complexity analysis of a discrete space-time cancer growth model with self-diffusion and cross-diffusion

Abstract

We investigate spatiotemporal pattern formation in cancer growth using discrete time and space variables. We first introduce the coupled map lattices (CMLs) model and provide a dynamical analysis of its fixed points along with stability results. We then offer parameter criteria for flip, Neimark–Sacker, and Turing bifurcations. In the presence of spatial diffusion, we find that stable homogeneous solutions can experience Turing instability under certain conditions. Numerical simulations reveal a variety of spatiotemporal patterns, including patches, spirals, and numerous other regular and irregular patterns. Compared to previous literature, our discrete model captures more complex and richer nonlinear dynamical behaviors, providing new insights into the formation of complex patterns in spatially extended discrete tumor models. These findings demonstrate the model’s ability to capture complex dynamics and offer valuable insights for understanding and treating cancer growth, highlighting its potential applications in biomedical research.

1 Introduction

Recently, the study of cancer growth models has attracted much attention. We hope to establish cancer growth mathematical models for theoretical analysis, and then control the growth of tumors and provide a theoretical basis for cancer treatment. And it is of great theoretical significance and practical application value to study the formation mechanism of spatio-temporal pattern from the perspective of mathematics [1–8]. It also allows us to understand the process of how immune cells and tumor cells interact with each other more intuitively in mathematical terms.

De Pillis and Radunskaya [9] proposed a three-dimensional cancer growth model, which is grounded in the Volterra–Lotka equation, which describes the interrelationship between normal cells, immune cells and tumor cells [10]. We consider that immune cells do not normally reply to tumor cells and that the activation mechanism in the immune system is dependent on the antigenic nature of the tumor. Thus, Itik [11] ignored the continuous influx for immune cells from the external environment. Some dynamic results of the model have been published in the paper, and previous studies have shown that the kinetics of the model are consistent with the results of clinical experiments [12]. Hence, it is highly meaningful to continue studying the model for control and treatment of cancer.

In this paper, we focus on examining a two-dimensional model in which tumor cells and immune cells compete with each other. In addition to the inherent cell death between the two, there exist diffusions between them in the organism, i.e., movement from high-density areas to low-density areas. In detail, we can view tumor demise as a predator-prey model. That is, tumor cells and immune cells play the roles of prey and predator, respectively [13, 14]. In order to avoid being engulfed by swarms of tumor cells, immune cells will move towards areas with low density of tumor cells, i.e., cross-diffusion. Cross-diffusion not only induces Turing destabilization, but also causes the cancer growth model to produce rich static patterns. Yet, it has been noted by a number of scientists that the Allee effect is also present in tumor growth [15, 16].

For the work on continuous cancer growth models, see [17–21] and the references in those. They studied models that were reaction-diffusion equations, whereas data collection in many cases is discontinuous in time, and the distribution of the biological populations is not spatially continuous [22–26]. In addition, algorithms for modeling spatial patterns are based on the corresponding discrete form of a continuous model. So discrete models are also significant to relate the real model to simulation. We find that Kaneko [27, 28] proposed a model of coupled mapping lattices (CMLs), which was later successfully used by Huang and Zhang [2] to explore the discrete spatio-temporal complexity of a predator-prey system. In contrast to the corresponding continuous system, CMLs models may simultaneously undergo spatio-temporal bifurcation, leading to four pattern formation mechanisms such as pure Turing instability, flip-Turing dynamics, Neimark–Sacker–Turing bifurcations, and chaotic oscillations. Surprisingly, CMLs models have richer dynamics and patterns than continuous systems [22, 29–34].

For CMLs models, several useful methods have been obtained, e.g., the Jacobian matrices employed for stabilization analysis, central manifold approximation [35], computation of spatial discretization operators [10], and the theoretical results on the Turing instability for a discrete system, which are available for scholars [31, 36–39]. One of the strengths of CMLs models is to keep the intrinsic characteristics of the initial system while being a more efficient method for numerical simulation [32, 33]. It also exhibits complex nonlinear features, e.g., freezing of stochastic patterns, spatial bifurcation, selection of patterns while suppressing chaos, chaotic defects exhibiting Brownian motion, turbulent anomalies, space-time interstices, completely developed space-time chaos, traveling waves, and super-transients [22, 27, 28, 30–33]. The CMLs method has been widely applied to chemical oscillators [29], ecosystems [33], neurodynamic systems [40], and many others.

Our research aims to explore bifurcations and chaotic phenomena in discrete systems and to compare these phenomena with corresponding results from continuous systems. Therefore, our primary approach involves using the CMLs method to study spatiotemporal dynamical behavior of a cancer growth model with strong Allee effect incorporating self-diffusion and cross-diffusion.

The paper is structured in the following manner. A description of CMLs models, discussions on the existence of fixed points, and stability results are included in Sect. 2. In Sect. 3, we study bifurcations of homogeneous stationary states, e.g., flip, Neimark–Sacker, and Turing bifurcations. Section 4 focuses on the numerical simulations of our theoretical results. Finally, Sect. 5 gives conclusions and summarizes whole paper.

2 CMLs models and stability analysis

Firstly, considering the relationship between the strong Allee effect of tumor cell growth and immune cells, we propose and study the mathematical model

$$ \textstyle\begin{cases} \dfrac{\partial T(t,x,y)}{\partial t} \\ \quad =r_{1} T(1-b_{1} T)(T-m) -c_{2} I T+d_{T} \nabla ^{2} T+d_{TI} \nabla ^{2} I, \quad t>0,\quad (x,y)\in \Omega , \\ \vspace{1mm} \dfrac{\partial I(t,x,y)}{\partial t}=\dfrac{\rho I T}{\alpha +T} -c_{1} I T-d_{1} I+d_{IT} \nabla ^{2} T+d_{I} \nabla ^{2} I,\quad t>0,\quad (x,y) \in \Omega , \\ \vspace{1mm} \dfrac{\partial T}{\partial \textbf{n}}= \dfrac{\partial I}{\partial \textbf{n}}=0,\qquad t>0,\quad (x,y)\in \Omega , \\ \vspace{1mm} T(0,x,y)\ge 0,\quad I(0,x,y)\ge 0,\qquad (x,y)\in \Omega =(0,L) \times (0,L), \end{cases} $$

(1)

where $T(t,x,y)$ is the amount of tumor cells at t moment and $I(t,x,y)$ is the amount of immune cells at t moment. $r_{1}$ is the average growth rate of tumor cells T, $\dfrac{1}{b_{1}}$ is the carrying capacity of tumor cells T, $c_{2}$ is the rate that the immune cells I kill the tumor cells T. $\dfrac{\rho I T}{\alpha +T}$ is a positive nonlinear term, ρ and α are positive constants; $c_{1}$ represents the rate of immune cells I inactivated by tumor cells T; $d_{1}$ is the average death rate of immune cells. m is the strong Allee threshold, where $0< m<\dfrac{1}{b_{1}}$. $d_{T}$ and $d_{I}$ represent the positive self-diffusion coefficients, $d_{TI}$ and $d_{IT}$ represent the cross-diffusion coefficients, where $\nabla ^{2}$ is the Laplacian operator in two dimensions. Ω is a bounded domain in $\mathbb{R}^{2}$ with the smooth boundary ∂Ω. n is the outward unit normal vector on ∂Ω and $\dfrac{1}{\partial \mathbf{n}}$ denotes the operator of the directional derivative along the direction n.

To approximate the model, the following is dimensionless: set $u=\dfrac{T}{\alpha}$, $v=\dfrac{c_{2}I}{\rho}$, $\overline{t}=\rho t$, $r=\dfrac{\alpha r_{1}}{\rho}$, $K=\dfrac{1}{\alpha b_{1}}$, $\overline{m}=\dfrac{m}{\alpha}$, $D_{1}=\dfrac{d_{T}}{\rho}$, $b=\dfrac{c_{1} \alpha}{\rho}$, $a=\dfrac{d_{1}}{\rho}$, $D_{2}=\dfrac{d_{I}}{\rho}$, $d_{12}=\dfrac{d_{TI}}{c_{2}\alpha}$, $d_{21}=\dfrac{c_{2} \alpha d_{IT}}{\rho ^{2}}$. For convenience, dropping the bar, then system (1) can be written as follows:

$$ \textstyle\begin{cases} \dfrac{\partial u(t,x,y)}{\partial t} \\ \quad =r u\left (1-\dfrac{u}{K}\right ) \left (u-m\right )-u v+D_{1} \nabla ^{2} u+d_{12} \nabla ^{2} v, \qquad t>0,\quad (x,y)\in \Omega , \\ \dfrac{\partial v(t,x,y)}{\partial t}=\dfrac{u v}{1+u}-b u v-a v+d_{21} \nabla ^{2} u+D_{2} \nabla ^{2} v,\qquad t>0,\quad (x,y)\in \Omega , \\ \vspace{1mm} \dfrac{\partial u}{\partial \textbf{n}}= \dfrac{\partial v}{\partial \textbf{n}}=0,\qquad t>0,\quad (x,y)\in \Omega , \\ \vspace{1mm} u(0,x,y)\ge 0,\quad v(0,x,y)\ge 0,\qquad (x,y)\in \Omega =(0,L) \times (0,L). \end{cases} $$

(2)

Then we construct models of CMLs and analyze their stability.

2.1 A cancer growth model for CMLs

On the basis of system (2), we establish the CMLs models. Within the two-dimensional matrix region, $n\times n$ grids are defined and time is discretized into a series of consecutive time periods. Using $u_{(i,j,t)}$ and $v_{(i,j,t)}$ ($i,j\in \{1,2,3,\ldots ,n\}$, $t\in \mathbb{Z^{+}}$) denote the number of tumor cells and immune cells on the $(i,j)$ grid at the time t between locations, where we assume that there is a local response and spatial diffusion [22, 32], and that the biomass of immune cells and tumor cells at each time point follow the dynamics of the system over time.

Between time t and $t+1$, the kinetic dynamics of cancer growth model and immune cells involve two successive steps: the “reaction” phase followed by the “diffusion” phase [2, 22, 32, 33]. The diffusion behavior precedes the reaction behavior. By introducing the space step δ and the time step τ, the spatial form of system (2) is discretised to obtain the equations controlling the diffusion process:

$$ \textstyle\begin{cases} u_{(i, j, t)}^{\prime}=u_{(i, j, t)}+\dfrac{\tau}{\delta ^{2}} D_{1} \nabla _{d}^{2} u_{(i, j, t)}+\dfrac{\tau}{\delta ^{2}} d_{12} \nabla _{d}^{2} v_{(i, j, t)}, \\ v_{(i, j, t)}^{\prime}=v_{(i, j, t)}+\dfrac{\tau}{\delta ^{2}} d_{21} \nabla _{d}^{2} u_{(i, j, t)}+\dfrac{\tau}{\delta ^{2}} D_{2} \nabla _{d}^{2} v_{(i, j, t)}, \end{cases} $$

(3)

where $u_{(i, j, t)}^{\prime}$ and $v_{(i, j, t)}^{\prime}$ represent the biomass of tumor cells and immune cells participating in the following reaction step. Laplacian operator $\nabla ^{2}$ is described in discrete form by $\nabla _{d}^{2}$:

$$ \textstyle\begin{cases} \nabla _{d}^{2} u_{(i, j, t)}=u_{(i+1, j, t)}+u_{(i-1, j, t)}+u_{(i, j+1, t)}+u_{(i, j-1, t)}-4 u_{(i, j, t)}, \\ \nabla _{d}^{2} v_{(i, j, t)}=v_{(i+1, j, t)}+v_{(i-1, j, t)}+v_{(i, j+1, t)}+v_{(i, j-1, t)}-4 v_{(i, j, t)}. \end{cases} $$

(4)

The discretization of Eq. (2) in the nonspatial domain yields the equations that control the reaction process:

$$ \textstyle\begin{cases} u_{(i, j, t+1)}=f(u_{(i, j, t)}^{\prime}, v_{(i, j, t)}^{\prime}), \\ v_{(i, j, t+1)}=g(u_{(i, j, t)}^{\prime}, v_{(i, j, t)}^{\prime}), \end{cases} $$

(5)

where

$$ \textstyle\begin{cases} f(u, v)=u+\tau \left (r u(1-\dfrac{u}{K})(u-m)-u v\right ), \\ g(u, v)=v+\tau \left (\dfrac{u v}{1+u}-b u v-a v\right ). \end{cases} $$

(6)

Eqs. (3)–(6) are CMLs models of system (2) with all parameters positive and $u_{(i,j,t)}\ge 0$ and $v_{(i,j,t)}\ge 0$. Then we study the dynamics of the CMLs models with the following periodic boundary conditions:

$$ \begin{aligned} & u_{(i, 0, t)}=u_{(i, n, t)},\quad u_{(i, 1, t)}=u_{(i, n+1, t)}, \\ & u_{(0, j, t)}=u_{(n, j, t)},\quad u_{(1, j, t)}=u_{(n+1, j, t)}, \\ & v_{(i, 0, t)}=v_{(i, n, t)},\quad v_{(i, 1, t)}=v_{(i, n+1, t)}, \\ & v_{(0, j, t)}=v_{(n, j, t)},\quad v_{(1, j, t)}=v_{(n+1, j, t)}. \end{aligned} $$

(7)

Cancer growth model in discrete time and space has spatially homogeneous and spatially heterogeneous dynamics. To each i, j, and t, spatially homogeneous dynamics requires that

$$\nabla _{d}^{2} u_{(i, j, t)} = 0, \quad \nabla _{d}^{2} v_{(i, j, t)} = 0. $$

(8)

Substituting into the CMLs models (3)–(6), homogeneous dynamics of a discrete system can be called

$$ \textstyle\begin{cases} u_{(i, j, t+1)}=f(u_{(i, j, t)}, v_{(i, j, t)})=u_{(i, j, t)}+\tau \left (r u_{(i, j, t)}(1-\dfrac{u_{(i, j, t)}}{K})(u_{(i, j, t)}-m)-u_{(i, j, t)} v_{(i, j, t)}\right ), \\ v_{(i, j, t+1)}=g(u_{(i, j, t)}, v_{(i, j, t)})=v_{(i, j, t)}+\tau ( \dfrac{u_{(i, j, t)} v_{(i, j, t)}}{1+u_{(i, j, t)}}-b u_{(i, j, t)} v_{(i, j, t)}-a v_{(i, j, t)}). \end{cases} $$

(9)

It may be written as follows:

$$ \left ( \textstyle\begin{array}{l} u \\ v \end{array}\displaystyle \right ) \mapsto \left ( \textstyle\begin{array}{l} u+\tau \left (r u(1-\dfrac{u}{K})(u-m)-u v\right ) \\ v+\tau (\dfrac{u v}{1+u}-b u v-a v) \end{array}\displaystyle \right ). $$

(10)

The homogeneous dynamics of the CMLs models (3)–(6) can be obtained by analyzing map (10). For spatially heterogeneous dynamics, it is necessary that there exists at least a set of i, j, t where $\nabla _{d}^{2} u_{(i, j, t)}\neq 0$ and $\nabla _{d}^{2} v_{(i, j, t)}\neq 0$.

2.2 Homogeneous steady state dynamics

Solve the following equations:

$$ \textstyle\begin{cases} u=f(u, v), \\ v=g(u, v), \end{cases} $$

we obtain the fixed points for map (10): $(u_{0},v_{0})=(0,0)$, $(u_{1},v_{1})=(K,0)$, $(u_{2},v_{2})=(m,0)$ and the positive fixed point $(u^{*},v^{*})$, where $u^{*}$ is the square root of the equation that follows:

$$ bu^{2}+(a+b-1)u+a=0, $$

correspondingly,

$$ v^{*}= \dfrac{r(K-u^{*})(u^{*}-m)}{K}. $$

Through direct calculation and analysis, we can determine the condition for the existence of positive fixed points for map (10). Map (10) has one and only one positive real fixed point, which we denote as $(u_{3},v_{3})$, where $u_{3}=\dfrac{-(a+b-1)-\sqrt{(a+b-1)^{2}-4ab}}{2b}$, $v_{3}=\dfrac{r(K-u_{3})(u_{3}-m)}{K}$, when the following condition is satisfied:

$$ (S1)\quad b\in (0,\dfrac{1}{(m+1)(K+1)}),\quad a\in ( \dfrac{-bm^{2}-m(b-1)}{m+1}, \dfrac{-bK^{2}-K(b-1)}{K+1}). $$

We give the fixed point stability theorem below.

Theorem 1

For the fixed points of map (10), we have:

• 1. When $0<\tau <\min \{\dfrac{2}{rm}, \dfrac{2}{a}\}$, $(u_{0},v_{0})=(0,0)$ is stable.

• 2. $(u_{1},v_{1})=(K,0)$ is stable when one of the following conditions is met:

  (H1) $a>a_{K}$, $\quad 0< b<\dfrac{1}{1+K}$, $\quad 0<\tau <\min \{\tau _{K}, \dfrac{-2}{r(m-K)}\}$;

  (H2) $a>0$, $\quad b>\dfrac{1}{1+K}$, $\quad 0<\tau <\min \{\tau _{K}, \dfrac{-2}{r(m-K)}\}$.

• 3. $(u_{2},v_{2})=(m,0)$ is always unstable.

• 4. $(u_{3},v_{3})$ is stable when one of the following conditions is met:

  (H3) $u_{3}>\dfrac{K+m}{2}$, $\quad 0<\tau <\tau ^{*}$;

  (H4) $u_{3}>\dfrac{K+m}{2}$, $\quad 0<\tau <\tau _{0}$.

  Further analysis shows that $(u_{3},v_{3})$ is a stable node when condition (H3) is satisfied, and $(u_{3},v_{3})$ is a stable focus when condition (H4) is satisfied,

where $a_{K}=\dfrac{-b K^{2}-b K+K}{1+K}$, $\tau _{K}=\dfrac{2(1+K)}{b K^{2}+(a+b-1) K+a}$, $\tau _{0}= \dfrac{r(K+m-2u_{3})(1+u_{3})^{2}}{Kv_{3}(b(1+u_{3})^{2}-1)}$, $\tau ^{*}=\tau _{0}-\sqrt{ \dfrac{(1+u_{3})^{2}\left ( \dfrac{r^{2}(K+m-2u_{3})^{2}(1+u_{3})^{2}}{K^{2}}+\dfrac{4(-1+b(1+u_{3})^{2}) v_{3}}{u_{3}}\right )}{(-1+b(1+u_{3})^{2})^{2} v_{3}^{2}}}$.

Proof

The Jacobian matrix for map (10) is

$$ J(u, v)=\left ( \textstyle\begin{array}{l@{\quad}c} 1+\tau \left (r\left (1-\dfrac{u}{K}\right )(u-m)-v+r u \dfrac{m+K-2 u}{K}\right ) & -u \tau \\ \tau \left (\dfrac{v}{(1+u)^{2}}-b v\right ) & 1+\tau (\dfrac{u}{1+u}-b u-a) \end{array}\displaystyle \right ). $$

For $(u_{0},v_{0})=(0,0)$, the corresponding Jacobian matrix is

$$ J(u_{0},v_{0})=\left ( \textstyle\begin{array}{c@{\quad}c} 1-\tau r m & 0 \\ 0 & 1-a \tau \end{array}\displaystyle \right ). $$

Clearly, the eigenvalues of $J(u_{0},v_{0})$ are $\lambda _{1}=-\tau r m+1$ and $\lambda _{2}=-a+1 \tau$. The absolute value of both $\lambda _{1}$ and $\lambda _{2}$ is less than 1 when $0<\tau <\min \{\dfrac{2}{rm}, \dfrac{2}{a}\}$ is satisfied, and the fixed point $(u_{0},v_{0})$ is stable [41].

For $(u_{1},v_{1})=(K,0)$, the corresponding Jacobian matrix is

$$ J(u_{1},v_{1})=\left ( \textstyle\begin{array}{c@{\quad}c} \tau r (m-K)+1 & - \tau K \\ 0 & \tau (\dfrac{K}{1+K}-b K-a)+1 \end{array}\displaystyle \right ). $$

Clearly, the eigenvalues of $J(u_{1},v_{1})$ are $\lambda _{1}=\tau r (m-K)+1$ and $\lambda _{2}=\tau (\dfrac{K}{1+K}-b K-a)+1$. When one of the following conditions is met:

$$ \textstyle\begin{cases} a>a_{K} \\ 0< b< \dfrac{1}{1+K} \\ 0< \tau < \min \{\tau _{K}, \dfrac{-2}{r(m-K)}\} \end{cases}\displaystyle ,\quad \textstyle\begin{cases} a>0 \\ b>\dfrac{1}{1+K} \\ 0< \tau < \min \{\tau _{K}, \dfrac{-2}{r(m-K)}\} \end{cases}\displaystyle , $$

the absolute value of both $\lambda _{1}$ and $\lambda _{2}$ is less than 1. The fixed points $(u_{1},v_{1})$ are stable.

Similarly, substituting $(u_{2},v_{2})=(m,0)$ into $J(u, v)$, we have

$$ J(u_{2},v_{2})=\left ( \textstyle\begin{array}{c@{\quad}c} 1+\tau rm \dfrac{K-m}{K} & -m \tau \\ 0 & 1+\tau (\dfrac{m}{1+m}-b m-a) \end{array}\displaystyle \right ). $$

Clearly, one of the eigenvalues of $J(u_{2},v_{2})$ is $1+\tau rm \dfrac{K-m}{K}>1$, so the fixed point $(u_{2},v_{2})$ is unstable.

Performing the same steps with the fixed point $(u_{3},v_{3})$, we get

$$ J(u_{3},v_{3})=\left ( \textstyle\begin{array}{l@{\quad}c} 1+\tau ru_{3} \dfrac{m+K-2u_{3}}{K} \quad & -u_{3} \tau \\ \tau \left (\dfrac{v_{3}}{(1+u_{3})^{2}}-bv_{3}\right ) \quad & 1 \end{array}\displaystyle \right ), $$

and the two eigenvalues of $J(u_{3},v_{3})$ as

$$ \lambda _{1,2}=\dfrac{1}{2}\left [\operatorname{Tr} J\left (u_{3},v_{3} \right ) \pm \sqrt{\left [\operatorname{Tr} J\left (u_{3},v_{3} \right )\right ]^{2}-4 \operatorname{Det} J\left (u_{3},v_{3}\right )} \right ], $$

where

$$\begin{aligned} &\operatorname{Tr}J\left (u_{3},v_{3}\right )=2+\tau r u_{3} \dfrac{m+K-2 u_{3}}{K}, \\ &\operatorname{Det}J\left (u_{3},v_{3}\right )=1+\tau r u_{3} \dfrac{m+K-2 u_{3}}{K}+\tau ^{2} u_{3} v_{3}\left ( \dfrac{1}{\left (1+u_{3}\right )^{2}}-b\right ). \end{aligned}$$

For $\left |\lambda _{1}\right |<1$ and $\left |\lambda _{2}\right |<1$, the stability conditions for $(u_{3},v_{3})$ are obtained as

$$ \operatorname{Det}J\left (u_{3},v_{3}\right )< 1,\quad -1- \operatorname{Det}J\left (u_{3},v_{3}\right ) < \operatorname{Tr}J \left (u_{3},v_{3}\right ) < 1+\operatorname{Det}J\left (u_{3},v_{3} \right ). $$

According to [41], we can directly compute the result. □

Based on the above stability analysis, we find that the discrete system may eventually converge to the homogeneous stationary states $(u_{0},v_{0})$, $(u_{1},v_{1})$, and $(u_{3},v_{3})$. While $(u_{0},v_{0})$ is a trivial stationary state, at $(u_{1},v_{1})$ the species v becomes extinct and only the species u exists, and then the interactions between u and v in terms of predation also disappear, which does not cause patterns of spatial heterogeneity to form. Therefore, only the production of bifurcations and patterns at $(u_{3},v_{3})$ will be considered in the following analysis.

3 Analysis of bifurcation in homogeneous stationary states

Next, we discuss bifurcation behaviors with τ as the major bifurcation parameter. Using this theoretical analysis, the parameter conditions supporting the formation of spatial patterns are obtained.

3.1 Flip bifurcation analysis

The stable node $(u_{3},v_{3})$ becomes unstable due to the flip bifurcation. At the same time, a new period 2 point appears. The first condition for a flip bifurcation is that the Jacobian matrix $J\left (u_{3},v_{3}\right )$ has $\lambda _{1}(\tau )=-1$ and $\left |\lambda _{2}(\tau )\right |\neq 1$ at the critical value. The following condition can satisfy this requirement:

$$\begin{aligned}& \tau =\tau ^{*}, \end{aligned}$$

(11)

$$\begin{aligned}& \tau ^{*} r u_{3}\dfrac{m+K-2 u_{3}}{K} \neq -2,-4. \end{aligned}$$

(12)

Next, the second condition for flip bifurcation can be obtained by the central manifold theorem. Take τ as the independent variable of map (10) and set $\bar{u}=u-u_{3}$, $\bar{v}=v-v_{3}$, and $\bar{\tau}=\tau -\tau ^{*}$, then we have

$$ \left ( \textstyle\begin{array}{l} \bar{u} \\ \bar{v} \\ \bar{\tau} \end{array}\displaystyle \right ) \rightarrow \left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c} a_{1} & a_{2} & 0 \\ b_{1} & b_{2} & 0 \\ 0 & 0 & 1 \end{array}\displaystyle \right )\left ( \textstyle\begin{array}{l} \bar{u} \\ \bar{v} \\ \bar{\tau} \end{array}\displaystyle \right )+\left ( \textstyle\begin{array}{l} f_{1}(\bar{u}, \bar{v}, \bar{\tau}) \\ f_{2}(\bar{u}, \bar{v}, \bar{\tau}) \\ f_{3}(\bar{u}, \bar{v}, \bar{\tau}) \end{array}\displaystyle \right ), $$

(13)

where

$$\begin{aligned}& \begin{aligned} f_{1}(\bar{u}, \bar{v}, \bar{\tau})&=\dfrac{1}{2} a_{3} \bar{u}^{2}+a_{4} \bar{u} \bar{v}+a_{5} \bar{u} \bar{\tau}+a_{6} \bar{v} \bar{\tau}+ \dfrac{1}{6} a_{7} \bar{u}^{3}+a_{8} \bar{u} \bar{v} \bar{\tau}+ \dfrac{1}{2} a_{9} \bar{u}^{2} \bar{\tau}+O(4), \\ f_{2}(\bar{u}, \bar{v}, \bar{\tau})&=\dfrac{1}{2} b_{3} \bar{u}^{2}+b_{4} \bar{u} \bar{v}+b_{5} \bar{u} \bar{\tau}+\dfrac{1}{6} b_{6} \bar{u}^{3}+b_{7} \bar{u} \bar{v} \bar{\tau}+\dfrac{1}{2} b_{8} \bar{u}^{2} \bar{v}+ \dfrac{1}{2} b_{9} \bar{u}^{2} \bar{\tau}+O(4), \\ f_{3}(\bar{u}, \bar{v}, \bar{\tau})&=0, \end{aligned} \\& \begin{aligned} & a_{1}=1+\tau ^{*} r u_{3} \dfrac{K+m-2 u_{3}}{K}, \quad a_{2}=-u_{3} \tau ^{*}, \quad a_{3}=\tau ^{*} r\left (2-\dfrac{6 u_{3}}{K}+ \dfrac{2 m}{K}\right ),\quad a_{4}=-\tau ^{*}, \\ &a_{5}=r u_{3} \dfrac{m+K-2 u_{3}}{K},\quad a_{6}=-u_{3},\quad a_{7}=- \dfrac{6}{K} r \tau ^{*},\quad a_{8}=-1, \\ &a_{9}=r\left (2-\dfrac{6}{K} u_{3}+\dfrac{2 m}{K}\right ), \\ &b_{1}=\tau ^{*}\left (\dfrac{v_{3}}{\left (1+u_{3}\right )^{2}}-b v_{3} \right ),\quad b_{2}=1,\quad b_{3}=\tau ^{*} \dfrac{-2 v_{3}}{\left (1+u_{3}\right )^{3}},\quad b_{4}=\tau ^{*} \left (\dfrac{1}{\left (1+u_{3}\right )^{2}}-b\right ), \\ & b_{5}=\dfrac{v_{3}}{\left (1+u_{3}\right )^{2}}-b v_{3},\quad b_{6}= \tau ^{*} \dfrac{b v_{3}}{\left (1+u_{3}\right )^{4}},\quad b_{7}= \dfrac{1}{\left (1+u_{3}\right )^{2}}-b,\quad b_{8}= \dfrac{-2 \tau ^{*}}{\left (1+u_{3}\right )^{3}}, \\ &b_{9}=\dfrac{-2 v_{3}}{\left (1+u_{3}\right )^{3}}. \end{aligned} \end{aligned}$$

(14)

The term $O(4)$ denotes polynomial functions of order higher than or equal to 4 in ū, v̄, and τ̄.

Apply the following inverse transformations to map (10):

$$ \left ( \textstyle\begin{array}{l} \bar{u} \\ \bar{v} \end{array}\displaystyle \right )=\left ( \textstyle\begin{array}{c@{\quad}c} a_{2} & a_{2} \\ -1-a_{1} & \lambda _{2}-a_{1} \end{array}\displaystyle \right )\left ( \textstyle\begin{array}{l} w \\ z \end{array}\displaystyle \right ), $$

(15)

then we can writesystem (13) as

$$ \left ( \textstyle\begin{array}{c} w \\ z \\ \bar{\tau} \end{array}\displaystyle \right ) \rightarrow \left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c} -1 & 0 & 0 \\ 0 & \lambda _{2} & 0 \\ 0 & 0 & 1 \end{array}\displaystyle \right )\left ( \textstyle\begin{array}{l} w \\ z \\ \bar{\tau} \end{array}\displaystyle \right )+\frac{1}{a_{2}\left (\lambda _{2}+1\right )}\left ( \textstyle\begin{array}{c} F_{1}(w, z, \bar{\tau}) \\ F_{2}(w, z, \bar{\tau}) \\ 0 \end{array}\displaystyle \right ), $$

(16)

where

$$\begin{aligned}& \begin{aligned} F_{1}(w, z, \bar{\tau}) =&\left (\lambda _{2}-a_{1}\right ) f_{1} \left (a_{2}(w+z),\left (-1-a_{1}\right ) w+\left (\lambda _{2}-a_{1} \right ) z, \bar{\tau}\right ) \\ & -a_{2} f_{2}\left (a_{2}(w+z),\left (-1-a_{1}\right ) w+\left ( \lambda _{2}-a_{1}\right ) z, \bar{\tau}\right ), \end{aligned} \\& \begin{aligned} F_{2}(w, z, \bar{\tau})=&\left (a_{1}+1\right ) f_{1}\left (a_{2}(w+z), \left (-1-a_{1}\right ) w+\left (\lambda _{2}-a_{1}\right ) z, \bar{\tau}\right ) \\ & +a_{2} f_{2}\left (a_{2}(w+z),\left (-1-a_{1}\right ) w+\left ( \lambda _{2}-a_{1}\right ) z, \bar{\tau}\right ). \end{aligned} \end{aligned}$$

Then, we analyze the center manifold $M^{c}(0,0,0)$ of map (16) at $(0,0,0)$ and use the center manifold theorem to show its existence and denote it as

$$ M^{c}(0,0,0)=\left \{(w, z, \bar{\tau}) \in R^{3} \mid z=g^{*}(w, \bar{\tau}), g^{*}(0,0)=0,Dg^{*}(0,0)=0\right \}, $$

(17)

where $g^{*}(w, \bar{\tau})$ is set to

$$ g^{*}(w, \bar{\tau})=e_{0} \bar{\tau}+e_{1} w^{2}+e_{2} w \bar{\tau}+e_{3} \bar{\tau}^{2}+O(3). $$

(18)

Substituting $z=g^{*}(w, \bar{\tau})$ into (16), we get

$$ \begin{aligned} &\lambda _{2} g^{*}(w, \bar{\tau})+ \dfrac{F_{2}\left (w, g^{*}(w, \bar{\tau}),\bar{\tau}\right )}{a_{2}\left (\lambda _{2}+1\right )} \\ &\quad = e_{0} \bar{\tau}+e_{1}\left [-w+ \dfrac{F_{1}\left (w, g^{*}(w, \bar{\tau}), \bar{\tau}\right )}{a_{2}\left (\lambda _{2}+1\right )} \right ]^{2} +e_{2}\left [-w+ \dfrac{F_{1}\left (w_{1} g^{*}\left (w, \bar{\tau}\right ), \bar{\tau}\right )}{a_{2}\left (\lambda _{2}+1\right )} \right ] \bar{\tau} \\ &\qquad {}+e_{3} \bar{\tau}^{2}+O(3). \end{aligned} $$

Comparing the two sides of τ̄, $w^{2}$, wτ̄, and $\bar{\tau}^{2}$, we have

$$ \begin{aligned} & e_{0}=0, \quad e_{1}= \dfrac{-2\left (1+a_{1}\right )^{2} a_{4}+a_{2}^{2} b_{3}+\left (1+a_{1}\right ) a_{2}\left (a_{3}-2 b_{4}\right )}{2\left (1-\lambda _{2}^{2}\right )}, \\ & e_{2}= \dfrac{\left (1+a_{1}\right )\left (-a_{2} a_{5}+a_{6}+a_{1} a_{6}\right )-a_{2}^{2} b_{5}}{a_{2}(1+\lambda _{2})^{2}}, \quad e_{3}=0. \end{aligned} $$

(19)

Correspondingly, in the limit of the central manifold, map (16) becomes

$$ F: \quad w \mapsto -w+\mu _{1} w^{2}+\mu _{2} w \bar{\tau}+\mu _{3} w^{2} \bar{\tau}+\mu _{4} w \bar{\tau}^{2}+u_{5} w^{3}+O(4), $$

(20)

where

$$ \begin{aligned} \mu _{1}={}&\dfrac{1}{2\left (1+\lambda _{2}\right )} \\ &{}\times \left (-a_{1} a_{2} a_{3}+2 a_{1} a_{4}+2 a_{1}^{2} a_{4}-a_{2}^{2} b_{3}+2\left (1+a_{1} \right ) a_{2} b_{4}+\left (a_{2} a_{3}-2\left (1+a_{1}\right ) a_{4} \right ) \lambda _{2}\right ), \\ \mu _{2}={}&\dfrac{1}{a_{2}\left (1+\lambda _{2}\right )}\left (-a_{2}^{2} b_{5}+\left (-a_{2} a_{5}+a_{6}+a_{1} a_{6}\right )\left (a_{1}- \lambda _{2}\right )\right ), \\ \mu _{3}={}&\dfrac{1}{a_{2}(1+\lambda _{2})} \\ &{}\times [a_{2}^{2} b_{7}+a_{1} a_{2}^{2} b_{7}-\dfrac{1}{2} a_{2}^{3} b_{9}-a_{2}^{2} b_{5} e_{1}-a_{2}^{3} b_{3} e_{2}+a_{2}^{2} b_{4} e_{2}+2 a_{1} a_{2}^{2} b_{4} e_{2}-a_{2}^{2} b_{4} e_{2}\lambda _{2} \\ & +\dfrac{1}{2}(a_{1}-\lambda _{2})[2 a_{1} a_{6} e_{1}-a_{2}^{2}(a_{9}+2 a_{3} e_{2})+2 a_{2}(a_{8}+a_{1} a_{8}-a_{5} e_{1}+a_{4} e_{2}+2 a_{1} a_{4} e_{2}) \\ & -2(a_{6} e_{1}+a_{2} a_{4} e_{2}) \lambda _{2}]], \\ \mu _{4}={}&\dfrac{e_{2}}{a_{2}\left (1+\lambda _{2}\right )}\left (-a_{1} a_{2} a_{5}+a_{1}^{2} a_{6}-a_{2}^{2} b_{5}+\lambda _{2}\left (a_{2} a_{5}-2 a_{1} a_{6}+a_{6} \lambda _{2}\right )\right ), \\ \mu _{5}={}&\dfrac{1}{6(1+\lambda _{2})} \\ &{}\times [-a_{2}^{2}(a_{1} a_{7}+a_{2} b_{6}-3(1+a_{1}) b_{8})+6(2 a_{1}^{2} a_{4}+a_{2}(-a_{2} b_{3}+b_{4})+a_{1}(-a_{2} a_{3}+a_{4} \\ & +2 a_{2} b_{4})) e_{1}+\lambda _{2}(a_{2}^{2} a_{7}-6(a_{4}+3 a_{1} a_{4}+a_{2}(-a_{3}+b_{4})) e_{1}+6 a_{4} e_{1} \lambda _{2})]. \end{aligned} $$

For map (20), the happening of the flip bifurcation requires two additional nonzero options [35]:

$$ \begin{aligned} & \eta _{1}=\left .\left ( \frac{\partial ^{2} F}{\partial w \partial \bar{\tau}}+\frac{1}{2} \frac{\partial F}{\partial \bar{\tau}} \frac{\partial ^{2} F}{\partial w^{2}}\right )\right |_{(w, \bar{\tau})=(0,0)}=\mu _{2} \neq 0, \\ & \eta _{2}=\left .\left (\frac{1}{6} \frac{\partial ^{3} F}{\partial w^{3}}+\left (\frac{1}{2} \frac{\partial ^{2} F}{\partial w^{2}}\right )^{2}\right )\right |_{(w, \bar{\tau})=(0,0)}=\mu _{5}+\mu _{1}^{2} \neq 0. \end{aligned} $$

An example is provided to illustrate the sign of $\eta _{2}$? in the parameter space $(a, b, \eta _{2})$. In Fig. 1(a), we show how $\eta _{2}$? varies with a and b. In Fig. 1(b), with b fixed at 0.01, we present how $\eta _{2}$? changes with a. Under this set of parameters, as a varies, $\eta _{2}$? is negative. Through the above analytical calculations, we can conclude the following.

Figure 1

(a) A surface $\eta _{2}$ within the parameter space $(a,b,\eta _{2})$; (b) The graph of the function $\eta _{2}$ within the parameter space $(a,\eta _{2})$ when $b=0.01$

Theorem 2

The CMLs models (3)–(6) undergo flip bifurcation if conditions (11) and (12) are satisfied with $\mu _{2} \neq 0$ and $\mu _{5} \neq -\mu _{1}^{2}$. When $\eta _{2}> 0$, the period 2 orbit diverging from the homogeneous steady state $(u_{3},v_{3})$ is stable; when $\eta _{2}< 0$, the period 2 orbit is unstable.

3.2 Neimark–Sacker bifurcation analysis

If there is a Neimark–Sacker bifurcation at the fixed point, it can bifurcate the invariant circle that bounds the fixed point. The Neimark–Sacker bifurcation requires that the two eigenvalues $\lambda _{1,2}$ of the corresponding Jacobian matrix $J\left (u_{3},v_{3}\right )$ are a pair of conjugate complex numbers, and $\left |\lambda _{1,2}\right |=1$. This implies that $\left [\operatorname{Tr} J\left (u_{3},v_{3}\right )\right ]^{2}-4 \operatorname{Det} J\left (u_{3},v_{3}\right )<0$, $\operatorname{Det} J\left (u_{3},v_{3}\right )=1$, namely,

$$\begin{aligned}& \tau =\tau _{0}, \end{aligned}$$

(21)

$$\begin{aligned}& \left (-2+ \dfrac{r^{2}\left (K+m-2 u_{3}\right )^{2} u_{3}}{K^{2}\left (-b v_{3}+\dfrac{v_{3}}{(1+u_{3})^{2}}\right )} \right )^{2}-4< 0. \end{aligned}$$

(22)

When the parameter conditions satisfy (21) and (22), we move the fixed point $(u_{3},v_{3})$ of map (10) to the origin:

$$ w=u-u_{3}, \quad z=v-v_{3}. $$

Then we have

$$ \begin{aligned} &w \mapsto a_{1} w+a_{2} z+\dfrac{1}{2} a_{3} w^{2}+a_{4} w z+ \dfrac{1}{6} a_{7} w^{3}+O(4), \\ &z \mapsto b_{1} w+b_{2} z+\dfrac{1}{2} b_{3} w^{2}+b_{4} w z+ \dfrac{1}{6} b_{6} w^{3}+\dfrac{1}{2} b_{8} w^{2} z+O(4). \end{aligned} $$

(23)

The coefficients $b_{1}$, $b_{2}$, $b_{3}$, $b_{4}$, $b_{6}$, $b_{8}$, $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $a_{7}$ are the same as previously defined in (14). However, $\tau ^{*}$ is replaced by $\tau _{0}$. Since translation does not affect the qualitative behaviors of fixed points, the Jacobian eigenvalues for map (23) at $(0,0)$ are conjugate complexes of mode 1, i.e.,

$$ \lambda \left (\tau _{0}\right ), \bar{\lambda}\left (\tau _{0} \right )=\dfrac{\operatorname{Tr} J\left (\tau _{0}\right )}{2} \pm \dfrac{\mathrm{i}}{2} \sqrt{4 \operatorname{Det}J\left (\tau _{0} \right )-\operatorname{Tr}^{2} J\left (\tau _{0}\right )}:=\alpha \pm \mathrm{i} \beta , $$

where $J\left (\tau _{0}\right )=J\left (u_{3}, v_{3}\right ) |_{\tau = \tau _{0}}$, $\mathrm{i}^{2}=1$, $\left |\lambda \left (\tau _{0}\right )\right | =\left | \bar{\lambda}\left (\tau _{0}\right )\right |=1$. Clearly, $|\lambda |=\sqrt{\operatorname{Det}J\left (\tau _{0}\right )}=1$, then we have

$$ d_{0}=\left .\dfrac{\mathrm{d}|\lambda (\tau )|}{\mathrm{d} \tau} \right |_{\tau =\tau _{0}} =\left .\dfrac{1}{2} \dfrac{\mathrm{d}(\mathrm{Det}J(\tau ))}{\mathrm{d} \tau}\right |_{ \tau =\tau _{0}} =\dfrac{-r u_{3}\left (m+K-2 u_{3}\right )}{2 K} \neq 0. $$

(24)

Furthermore, the two eigenvalues of the Neimark–Sacker bifurcation are not real nor imaginary. Then

$$ \left (\lambda \left (\tau _{0}\right )\right )^{\theta }\neq 1, \quad \theta =1,2,3,4, $$

(25)

namely,

$$ \tau _{0} r u_{3} \dfrac{m+K-2 u_{3}}{K} \neq -2,-3. $$

(26)

Next, we require the type of normal to map (23) via the central manifold approximation to obtain the last determinant of the Neimark–Sacker bifurcation. Assuming that

$$ \left ( \textstyle\begin{array}{l} w \\ z \end{array}\displaystyle \right )=\left ( \textstyle\begin{array}{c@{\quad}c} a_{2} & 0 \\ \alpha -a_{1} & -\beta \end{array}\displaystyle \right )\left ( \textstyle\begin{array}{l} \widetilde{w} \\ \widetilde{z} \end{array}\displaystyle \right ), $$

map (23) is written as

$$ \begin{aligned} & \left ( \textstyle\begin{array}{c} \widetilde{w} \\ \widetilde{z} \end{array}\displaystyle \right ) \mapsto \left ( \textstyle\begin{array}{l@{\quad}l} \alpha & -\beta \\ \beta & \alpha \end{array}\displaystyle \right )\left ( \textstyle\begin{array}{l} \widetilde{w} \\ \widetilde{z} \end{array}\displaystyle \right )+\frac{1}{a_{2} \beta}\left ( \textstyle\begin{array}{l} G_{1}(\widetilde{w}, \tilde{z}) \\ G_{2}(\widetilde{w}, \tilde{z}) \end{array}\displaystyle \right ), \end{aligned} $$

(27)

where

$$ \begin{aligned} G_{1}(\widetilde{w}, \widetilde{z})=&-a_{2} a_{4} \beta ^{2} \widetilde{w}\widetilde{z}+(\dfrac{1}{2} a_{2}^{2} a_{3} \beta -a_{1} a_{2} a_{4} \beta +a_{4} a_{2} \alpha \beta ) \widetilde{w}^{2}+ \dfrac{1}{6} a_{2}^{3} a_{7} \beta \widetilde{w}^{3}+O(4), \\ G_{2}(\widetilde{w}, \widetilde{z})=&a_{2}\left (a_{1} a_{4}+a_{2} b_{4}-a_{4} \alpha \right ) \beta \widetilde{w} \widetilde{z} +\dfrac{1}{2}a_{2}(2 a_{1}^{2} a_{4}+a_{2} \alpha \left (a_{3}-2 b_{4}\right )+2 a_{4} \alpha ^{2} \\ &-a_{1}\left [a_{2}\left (a_{3}-2 b_{4}\right )+4 \alpha a_{4}\right ]-a_{2}^{2} b_{3}) \widetilde{w}^{2}+\dfrac{1}{2} a_{2}^{3} b_{8} \beta \widetilde{w}^{2} \widetilde{z}-\dfrac{1}{6} a_{2}^{3} \widetilde{w}^{3}[a_{2} b_{6} \\ &+a_{1}\left (a_{7}-3 b_{8}\right )-a_{7} \alpha +3 b_{8} \alpha ]+O(4). \end{aligned} $$

At $(\widetilde{w}, \widetilde{z})=(0,0)$, we have

$$ \begin{aligned} & G_{1 \widetilde{w} \widetilde{w}}=\dfrac{1}{2} a_{2}^{2} a_{3} \beta -a_{1} a_{2} a_{4} \beta +a_{4} a_{2} \alpha \beta ,\quad G_{1 \widetilde{w} \widetilde{z}}=-a_{2} a_{4} \beta ^{2},\quad G_{1 \widetilde{z} \widetilde{z}}=0, \\ &G_{1 \widetilde{w} \widetilde{w} \widetilde{w}}=\dfrac{1}{6} a_{2}^{3} a_{7} \beta , \\ &G_{1 \widetilde{w} \widetilde{w} \widetilde{z}}=G_{1 \widetilde{w} \widetilde{z} \widetilde{z}}=G_{1 \widetilde{z} \widetilde{z} \widetilde{z}}=0,\quad G_{2 \widetilde{w} \widetilde{z}}=a_{2}\left (a_{1} a_{4}+a_{2} b_{4}-a_{4} \alpha \right ) \beta ,\quad G_{2 \widetilde{z} \widetilde{z}}=0, \\ & G_{2 \widetilde{w} \widetilde{w}}=\dfrac{1}{2} a_{2}\left [2 a_{1}^{2} a_{4}+a_{2} \alpha \left (a_{3}-2 b_{4}\right )+2 a_{4} \alpha ^{2}-a_{1} \left (a_{2}\left (a_{3}-2 b_{4}\right )+4 \alpha a_{4}\right )-a_{2}^{2} b_{3}\right ], \\ & G_{2 \widetilde{w} \widetilde{w} \widetilde{w}}=-\dfrac{1}{6} a_{2}^{3} \left [a_{2} b_{6}+a_{1}\left (a_{7}-3 b_{8}\right )-a_{7} \alpha +3 b_{8} \alpha \right ],\quad G_{2 \widetilde{w} \widetilde{w} \widetilde{z}}= \dfrac{1}{2} a_{2}^{3} b_{8} \beta , \\ &G_{2 \widetilde{w} \widetilde{z} \widetilde{z}}=G_{2 \widetilde{z} \widetilde{z} \widetilde{z}}=0. \end{aligned} $$

In order to make sure that the Neimark–Sacker bifurcation of map (27) occurs, we require that the determinant σ satisfies

$$ \sigma =-\operatorname{Re}\left [ \dfrac{(1-2 \lambda ) \bar{\lambda}^{2}}{1-\lambda} \xi _{11} \xi _{20} \right ]-\dfrac{1}{2}\left |\xi _{11}\right |^{2}-\left |\xi _{02} \right |^{2}+\operatorname{Re}\left (\bar{\lambda} \xi _{21}\right ) \neq 0, $$

(28)

where

$$\begin{aligned}& \begin{aligned} \xi _{20}=&\dfrac{1}{8 a_{2} \beta}[G_{1 \widetilde{w} \widetilde{w}}-G_{1 \widetilde{z} \widetilde{z}}+2 G_{2 \widetilde{w} \widetilde{z}} + \mathrm{i}(G_{2 \widetilde{w} \widetilde{w}}-G_{2 \widetilde{z} \widetilde{z}}-2 G_{1 \widetilde{w} \widetilde{z}})] \\ =&\dfrac{1}{16 \beta}[\beta (a_{2}(a_{3}+4 b_{4})+2 a_{4}(a_{1}- \alpha ))+\mathrm{i}(2 a_{1}^{2} a_{4}-a_{2}^{2} b_{3}+a_{2}(a_{3}-2 b_{4}) \alpha \\ &-a_{1}(a_{2}(a_{3}-2 b_{4})+4 a_{4}\alpha )+2 a_{4}(\alpha ^{2}+2 \beta ^{2}))], \end{aligned} \\& \begin{aligned} \xi _{11}={}&\dfrac{1}{4 a_{2} \beta}\left [G_{1 \widetilde{w} \widetilde{w}}+G_{1 \widetilde{z} \widetilde{z}}+\mathrm{i}\left (G_{2 \widetilde{w} \widetilde{w}}+G_{2 \widetilde{z} \widetilde{z}}\right ) \right ] \\ ={}&\dfrac{1}{8 a_{2} \beta}[a_{2} \beta (a_{2} a_{3}+2a_{4}(\alpha -a_{1}))+ \mathrm{i} a_{2}(2 a_{1}^{2} a_{4}-a_{2}^{2} b_{3}+a_{2}(a_{3}-2 b_{4}) \alpha +2 a_{4} \alpha ^{2} \\ &{}-a_{1}(a_{2}(a_{3}-2 b_{4})+4 \alpha a_{4}))], \\ \xi _{02}=&\dfrac{1}{8 a_{2} \beta}[G_{1 \widetilde{w} \widetilde{w}}-G_{1 \widetilde{z}\widetilde{z}}-2 G_{2 \widetilde{w} \widetilde{z}}+ \mathrm{i}(G_{2 \widetilde{w} \widetilde{w}}-G_{2\widetilde{z} \widetilde{z}}+2 G_{1 \widetilde{w} \widetilde{z}})] \\ ={}&\dfrac{1}{8a_{2} \beta}[\dfrac{1}{2} a_{2} \beta (a_{2}(a_{3}-4 b_{4})+6 a_{4}(\alpha -a_{1}))-\mathrm{i}\dfrac{1}{2} a_{2}(-2 a_{1}^{2} a_{4}+a_{2}^{2} b_{3}-a_{2} a_{3} \alpha \\ &{} +2 a_{2} b_{4} \alpha -2 a_{4} \alpha ^{2}+a_{1}(a_{2}(a_{3}-2 b_{4})+4 \alpha a_{4})+4 a_{4} \beta ^{2})], \\ \xi _{21}=&\dfrac{1}{16 a_{2} \beta}\left [G_{1 \widetilde{w} \widetilde{w}\widetilde{w}}+G_{1\widetilde{w} \widetilde{z} \widetilde{z}}+G_{2\widetilde{w} \widetilde{w} \widetilde{z}}+G_{2 \widetilde{z} \widetilde{z} \widetilde{z}}+\mathrm{i}\left (G_{2 \widetilde{w} \widetilde{w}\widetilde{w}}+G_{2 \widetilde{w} \widetilde{z} \widetilde{z}}-G_{1\widetilde{w} \widetilde{w} \widetilde{z}}-G_{1\widetilde{z} \widetilde{z} \widetilde{z}}\right ) \right ] \\ ={}&\dfrac{1}{96} a_{2}^{2}\left (a_{7}+3 b_{8}\right )- \dfrac{a_{2}^{2}}{96 \beta}\mathrm{i}\left (a_{2} b_{6}+\left (a_{7}-3 b_{8}\right )\left (a_{1}-\alpha \right )\right ). \end{aligned} \end{aligned}$$

Next, we provide an example to illustrate the sign of σ in the parameter space $(a,b,\sigma )$. In Fig. 2(a), we show how σ varies with a and b. In Fig. 2(b), with b fixed at 0.01, we present how σ changes with a. Under this set of parameters, as a varies, σ can be either positive or negative. The following results are obtained from the above analysis and calculations.

Figure 2

(a) A surface σ within the parameter space $(a,b,\sigma )$; (b) The graph of the function σ within the parameter space $(a,\sigma )$ when $b=0.01$

Theorem 3

If conditions (21), (22), (24), (26), (28) are satisfied, then the CMLs models (3)–(6) occur Neimark–Sacker bifurcation at the fixed point $(u_{3},v_{3})$. In addition, when $\sigma <0$ and $d_{0}>0$, for $\tau >\tau _{0}$, an invariant circle of attraction bifurcates at $(u_{3},v_{3})$, and when $\sigma >0$ and $d_{0}>0$, for $0<\tau <\tau _{0}$, an invariant circle of repulsion bifurcates at $(u_{3},v_{3})$.

3.3 Turing bifurcation analysis

Spatial symmetry destruction is the major reason why Turing bifurcation occurs. Due to the inhomogeneity of spatial diffusion, the homogeneous steady state of CMLs models transitions into an unstable state if Turing instability occurs. There are two necessary conditions for Turing bifurcation to occur [2]. Firstly, nontrivial homogeneous steady states are stable in time. Secondly, stable nontrivial homogeneous steady states are unstable in space under heterogeneous perturbations. By Theorem 1, $(u_{3},v_{3})$ is stable in time if either condition $(H3)$ or $(H4)$ is satisfied.

Next, we consider that either condition $(H3)$ or $(H4)$ stands and discuss the Turing bifurcation at homogeneous steady state. To obtain conditions for Turing instability, we consider

$$ \nabla _{d}^{2} X^{i j}+\lambda X^{i j}=0, $$

(29)

satisfying the periodic boundary conditions

$$ X^{i, 0}=X^{i, n},\quad X^{i, 1}=X^{i, n+1},\quad X^{0, j}=X^{n, j}, \quad X^{1, j}=X^{n+1, j}. $$

From [42], we get that the eigenvalue $\nabla _{d}^{2}$ of $\lambda _{kl}$ satisfies

$$ \lambda _{k l}=4\left [\sin ^{2} \dfrac{(k-1) \pi}{n}+\sin ^{2} \dfrac{(l-1) \pi}{n}\right ] \triangleq 4\left (\sin ^{2} \phi _{k}+ \sin ^{2} \phi _{l}\right ), $$

with $k,l\in \{1,2,3,\ldots ,n\}$. The eigenvalue $\lambda _{kl}$ corresponding to the eigenfunction is denoted as $X_{kl}^{ij}$, i.e.,

$$ \nabla _{d}^{2} X_{kl}^{ij}+\lambda _{kl} X_{kl}^{ij}=0. $$

Next, we study the stability of $(u_{3},v_{3})$ under small-space heterogeneous perturbations. Let

$$ \widetilde{u}_{(i, j,t)}= u_{(i, j, t)}-u_{3}, \quad \widetilde{v}_{(i, j, t)}=v_{(i, j, t)}-v_{3}. $$

Since $\nabla _{d}^{2} \widetilde{u}_{(i, j, t)}=\nabla _{d}^{2} u_{(i j, t)}$ and $\nabla _{d}^{2} \widetilde{v}_{(i, j, t)}=\nabla _{d}^{2} v_{(i, j, t)}$, we have

$$ \textstyle\begin{cases} \begin{aligned} \widetilde{u}_{(i, j, t+1)}=&a_{1}\left (\widetilde{u}_{(i, j, t)}+ \dfrac{\tau}{\delta ^{2}} D_{1} \nabla _{d}^{2} \widetilde{u}_{(i, j, t)}+ \dfrac{\tau}{\delta ^{2}} d_{12} \nabla _{d}^{2} \widetilde{v}_{(i, j, t)}\right ) \\ &+a_{2}\left (\widetilde{v}_{(i, j, t)}+\dfrac{\tau}{\delta ^{2}} d_{21} \nabla _{d}^{2} \widetilde{u}_{(i, j, t)}+\dfrac{\tau}{\delta ^{2}} D_{2} \nabla _{d}^{2} \widetilde{v}_{(i, j, t)}\right )+O\left ((| \widetilde{u}_{(i, j, t)}|+|\widetilde{v}_{(i, j, t)}|)^{2}\right ), \\ \widetilde{v}_{(i, j, t+1)}=&b_{1}\left (\widetilde{u}_{(i, j, t)}+ \dfrac{\tau}{\delta ^{2}} D_{1} \nabla _{d}^{2} \widetilde{u}_{(i, j, t)}+ \dfrac{\tau}{\delta ^{2}} d_{12} \nabla _{d}^{2} \widetilde{v}_{(i, j, t)}\right ) \\ &+b_{2}\left (\widetilde{v}_{(i, j, t)}+\dfrac{\tau}{\delta ^{2}} d_{21} \nabla _{d}^{2} \widetilde{u}_{(i, j, t)}+\dfrac{\tau}{\delta ^{2}} D_{2} \nabla _{d}^{2} \widetilde{v}_{(i j, t)}\right )+O\left ((| \widetilde{u}_{(i, j, t)}|+|\widetilde{v}_{(i, j, t)}|)^{2}\right ), \end{aligned} \end{cases} $$

(30)

where $a_{1}$, $a_{2}$, $b_{1}$, $b_{2}$ are defined in (14). The linear term predominates the dynamics of system (30) when the perturbations are small. We multiply each end of equations (30) by $X_{kl}^{ij}$ to get

$$ \textstyle\begin{cases} \begin{aligned} X_{kl}^{ij}\widetilde{u}_{(i, j, t+1)}=&X_{kl}^{ij}\left (a _{1} \widetilde{u}_{(i, j, t)}+a_{2}\widetilde{v}_{(i, j, t)}\right )+ \dfrac{\tau}{\delta ^{2}}X_{kl}^{ij}\left (a_{1} D_{1} \nabla _{d}^{2} \widetilde{u}_{(i, j, t)}+a_{1}d_{12}\nabla _{d}^{2} \widetilde{u}_{(i, j, t)}\right . \\ &\left .+a_{2}d_{21} \nabla _{d}^{2} \widetilde{u}_{(i, j, t)} +a_{2}D_{2} \nabla _{d}^{2} \widetilde{v}_{(i, j, t)}\right ), \\ X_{kl}^{ij}\widetilde{v}_{(i, j, t+1)}=&X_{kl}^{ij}\left (b_{1} \widetilde{u}_{(i, j, t)}+b_{2}\widetilde{v}_{(i, j, t)}\right )+ \dfrac{\tau}{\delta ^{2}}X_{kl}^{ij}\left (b_{1} D_{1} \nabla _{d}^{2} \widetilde{u}_{(i, j, t)}+b_{1}d_{12}\nabla _{d}^{2} \widetilde{u}_{(i, j, t)}\right . \\ &\left .+b_{2}d_{21} \nabla _{d}^{2} \widetilde{u}_{(i, j, t)} +b_{2}D_{2} \nabla _{d}^{2} \widetilde{v}_{(i, j, t)}\right ). \end{aligned} \end{cases} $$

(31)

We sum all i and j of the above equations to obtain

$$ \textstyle\begin{cases} \sum _{i, j=1}^{n}X_{kl}^{ij}\widetilde{u}_{(i, j, t+1)} \\ \quad =a_{1}\sum _{i, j=1}^{n}X_{kl}^{ij}\widetilde{u}_{(i, j, t)}+a_{2}\sum _{i, j=1}^{n}X_{kl}^{ij} \widetilde{v}_{(i, j, t)}+\dfrac{\tau}{\delta ^{2}}(a_{1} D_{1}+a_{2}d_{21}) \sum _{i, j=1}^{n}X_{kl}^{ij} \nabla _{d}^{2} \widetilde{u}_{(i, j, t)} \\ \qquad {}+\dfrac{\tau}{\delta ^{2}}(a_{2}D_{2}+a_{1}d_{12}) \sum _{i, j=1}^{n}X_{kl}^{ij} \widetilde{v}_{(i, j, t)}, \\ \sum _{i, j=1}^{n}X_{kl}^{ij}\widetilde{v}_{(i, j, t+1)} \\ \quad =b_{1}\sum _{i, j=1}^{n}X_{kl}^{ij}\widetilde{u}_{(i, j, t)}+b_{2}\sum _{i, j=1}^{n}X_{kl}^{ij} \widetilde{v}_{(i, j, t)}+\dfrac{\tau}{\delta ^{2}}(b_{1} D_{1}+b_{2}d_{21}) \sum _{i, j=1}^{n}X_{kl}^{ij} \nabla _{d}^{2} \widetilde{u}_{(i, j, t)} \\ \qquad {}+\dfrac{\tau}{\delta ^{2}}(b_{2}D_{2}+b_{1}d_{12}) \sum _{i, j=1}^{n}X_{kl}^{ij} \widetilde{v}_{(i, j, t)}. \end{cases} $$

(32)

Suppose $\bar{u}_{t}=\sum _{i, j=1}^{n}X_{kl}^{ij}\widetilde{u}_{(i, j, t)}$ and $\bar{v}_{t}=\sum _{i, j=1}^{n}X_{kl}^{ij}\widetilde{v}_{(i, j, t)}$, then system (32) may be written in the following way:

$$ \textstyle\begin{cases} \bar{u}_{t+1}=\left (a_{1}-\dfrac{\tau}{\delta ^{2}}(a_{1} D_{1}+a_{2} d_{21}) \lambda _{k l}\right ) \bar{u}_{t}+\left (a_{2}- \dfrac{\tau}{\delta ^{2}} (a_{1}d_{12}+a_{2}D_{2}) \lambda _{k l} \right ) \bar{v}_{t}, \\ \bar{v}_{t+1}=\left (b_{1}-\dfrac{\tau}{\delta ^{2}}(b_{1} D_{1}+b_{2} d_{21}) \lambda _{k l}\right ) \bar{u}_{t}+\left (b_{2}- \dfrac{\tau}{\delta ^{2}} (b_{1}d_{12}+b_{2}D_{2}) \lambda _{k l} \right ) \bar{v}_{t}. \end{cases} $$

(33)

System (33) describes the dynamical sum of spatially inhomogeneous perturbations on all discrete lattices of the entire two-dimensional space. The spatially homogeneous steady state $(u_{3},v_{3})$ of the CMLs models (3)–(6) is stable when system (33) is stable at origin, otherwise $(u_{3},v_{3})$ is unstable, and the latter results in Turing instability.

We want to know the condition that at least one of the eigenvalues has mode larger than 1. At $(0,0)$, the Jacobian matrix of system (33) is

$$ \left ( \textstyle\begin{array}{l@{\quad}l} a_{1}-\dfrac{\tau}{\delta ^{2}}(a_{1} D_{1}+a_{2} d_{21}) \lambda _{k l} &\quad a_{2}-\dfrac{\tau}{\delta ^{2}} (a_{1}d_{12}+a_{2}D_{2}) \lambda _{k l} \\ b_{1}-\dfrac{\tau}{\delta ^{2}}\left (b_{1} D_{1}+b_{2} d_{21}\right ) \lambda _{k l} &\quad b_{2}-\dfrac{\tau}{\delta ^{2}} (b_{1}d_{12}+b_{2}D_{2}) \lambda _{k l} \end{array}\displaystyle \right ), $$

and the respective eigenvalues are

$$ \lambda _{ \pm}(k, l, \tau )=\dfrac{1}{2} R(k, l, \tau ) \pm \dfrac{1}{2} \sqrt{R^{2}(k, l, \tau )-4 Q(k, l, \tau )}, $$

(34)

with

$$ \begin{aligned} R(k, l, \tau ) & =a_{1}+b_{2}-\dfrac{\tau}{\delta ^{2}} \lambda _{k l}(a_{1} D_{1}+a_{2} d_{21}+b_{1}d_{12}+b_{2} D_{2}), \\ Q(k, l, \tau ) & =\dfrac{1}{\delta ^{4}}(a_{1} b_{2}-a_{2} b_{1}) \left (\delta ^{4}-(D_{1}+D_{2})\delta ^{2} \lambda _{k l} \tau +(D_{1}D_{2}-d_{12}d_{21}) \lambda _{k l}^{2}\tau ^{2}\right ). \end{aligned} $$

(35)

Set

$$ \begin{aligned} & Z(k, l, \tau )=\max \left \{\left |\lambda _{ \pm}(k, l, \tau ) \right |\right \}, \\ & Z_{m}(\tau )=\max _{k, l=1}^{n} Z(k, l, \tau ), \quad \big((k, \tau ) \neq (1,1)\big). \end{aligned} $$

(36)

The critical value of Turing bifurcation $\tau ^{\prime}$ is obtained by solving the equation $Z_{m}(\tau )=1$. Furthermore, as τ approaches $\tau ^{\prime}$, if $R^{2}(k,l,\tau )>4Q(k,l,\tau )$, then $\tau ^{\prime}$ satisfies the following condition:

$$ \max _{k=1, l=1}^{n}\left \{\left |R\left (k, l, \tau ^{\prime} \right )\right |-Q\left (k, l, \tau ^{\prime}\right )\right \}=1. $$

(37)

If $R^{2}(k, l, \tau )\le 4Q(k, l, \tau )$, then $\tau ^{\prime}$ is satisfied by

$$ \max _{k=1, l=1}^{n} Q\left (k, l, \tau ^{\prime}\right )=1. $$

(38)

The above calculation shows the theorem below.

Theorem 4

If one of conditions $(H3)$ or $(H4)$ of Theorem 1holds and τ lie within the range of $\tau ^{\prime}$, the homogeneous steady state of the CMLs models (3)–(6) subjected to the periodic boundary condition (7) when $Z_{m}(\tau )>1$ undergoes Turing bifurcation. Hence, Turing patterns are formed. The homogeneous steady state $(u_{3},v_{3})$ is stable with $Z_{m}(\tau )<1$ and without Turing pattern.

Summarizing, the mechanism by which the induced patterns occur depends on the type of bifurcations of the homogeneous steady state.

4 Numerical simulation

In this part, we perform numerical simulations to show the dynamical evolution and spatiotemporal pattern.

4.1 Flip bifurcation and its corresponding patterns

We set $r=10$, $m=4.5$, $K=5$, $b=0.01$, $a=0.78$ and compute the rectangular grid as $n=200$.

Clearly, the fixed point is $(u_{3},v_{3})=(4.8211,0.1149)$ and the threshold value for the flip bifurcation is $\tau ^{*} = 1.4673208$. Setting $\tau = \tau ^{*}$, the eigenvalues are −1 and 0.988366, $\eta _{1} = -1.36303 < 0$, and $\eta _{2} = -402.841 < 0$. From Theorem 3, the orbits of the bifurcation period 2 are unstable if τ near $\tau ^{*}$. In Fig. 3(a), we draw the associated bifurcation at $\tau \in [1.4,1.6]$. The original value of the flip bifurcation is $(u_{3}+ 0.0001,v_{3}+ 0.0001)$. We see exactly the period-folding cascade of u from Fig. 3(a). For more results from the cycle window, local amplification of $\tau \in [1.55,1.59]$ is plotted in Fig. 3(b). Within the period window, a period 6 orbit can be seen.

Figure 3

(a) Flip bifurcation diagram for map (10) when $\tau \in [1.4,1.6]$; (b) local amplification of Fig. 3(a) with $\tau \in [1.55,1.59]$

We plot the maximum Lyapunov exponent in Figure 4 corresponding to Fig. 3(a), it helps us quantify whether chaotic behavior is occurring or not. According to Fig. 4(b), maximum Lyapunov exponent is greater than zero at τ about 1.583. This suggests that chaotic behavior may be happening.

Figure 4

(a) Maximum Lyapunov exponent corresponding to Fig. 3(a); (b) local amplification of Fig. 4(a)

Below, we will focus on the dynamic transformation of map (10) as τ becomes larger. In Fig. 5(a), we can get a period 1 orbit. Let $\tau =1.46$, $\tau =1.47$, $\tau =1.489$, and $\tau =1.491$, we get orbits with periods 2, 4, 8, and 16, respectively, see Figs. 5(b)–(e). For $\tau =1.492$, we get orbits with different periods, see Fig. 5(f). Then, we let $\tau = 1.571$ to obtain the period 6 orbits that correspond to the period window in Fig. 3(b), please see Fig. 5(g). Finally, we can see the chaotic attractor with $\tau =1.583$ in Fig. 5(h). From Fig. 5, we can clearly see the dynamics from stable fixed points to chaotic paths for different τ.

Figure 5

Phase diagram varying with the value of parameter τ

Next, we identify threshold value $\tau ^{\prime }$ using the Turing bifurcation diagram. The regions where the pattern is formed are then plotted. Depending on the region, we show that the patterns transform.

Set $D_{1}=0.03$, $D_{2}=1$, $d_{12}=0$, $d_{21}=0$, $\delta =10$, and plot the $Z_{m}-\tau$ diagram in Fig. 6(a). Clearly, the threshold for Turing bifurcation is $\tau ^{\prime }\approx 1.466325$. Together with the Turing bifurcation curve $\tau =\tau ^{\prime }$ and the flip bifurcation curve $\tau =\tau ^{*}$, we obtain the region of pattern formation in Fig. 6(b) where $D_{2}$ varies between 0 and 10. There are three regions: the homogeneous steady state, the pure Turing instability, and the flip-Turing instability regions.

Figure 6

(a) $Z_{m}-\tau$ diagram of the Turing bifurcation; (b) $\tau -D_{2}$ diagram shows regions of patterns formation

Below, we show the patterns induced by self-diffusion. Setting $D_{1}=0.03$, $D_{2}=1$, $d_{12}=0$, $d_{21}=0$, and $\delta =10$, we simulate the spatial patterns of biomass u caused by flip-Turing instability and chaotic mechanisms relative to Fig. 5, as shown in Fig. 7. The spatial patterns in the figures are all spatial distributions at $t = 20000$. The initial state is the homogenous steady state $(u_{3},v_{3})$ with random perturbations.

Figure 7

On the $200\times 200$ lattices, $D_{1} = 0.03$, $D_{2} = 1$, $d_{12} = 0$, $d_{21} = 0$, and $\delta = 10$, and the spatial patterns caused under flip-Turing instability and chaos for an iteration step size of 20000

When $\tau =1.45$, both the Turing bifurcation and the flip bifurcation do not occur, and the homogeneous stationary state is stable. Thus, there is no formation of spatial patterns, see Fig. 7(a). When $\tau =1.47$, the CMLs models (3)–(6) undergo both Turing and flip bifurcations. This is when the CMLs model forms spatially heterogeneous patterns that are arising from the flip-Turing instability, and we will find that the four states of u are intertwined with each other, and the patterns consist of four colors representing the period 4 points. Please refer to Fig. 5(b). Assuming $\tau = 1.489$, we observe an 8-state mosaic of spatial patterns dominated by the period 8 points of u (Fig. 7(c)). The pattern when $\tau =1.491$ and so on. Hence, we do not describe them. When $\tau =1.492$, we see more fragmentary patterns as Fig. 7(e). When $\tau =1.583$, we see that the CMLs models are chaotic according to Fig. 5(h). At the same time, the associated patterns also exhibit chaotic features. The patterns are tessellated. We do not know how many colors are in the patterns. From the above simulation, we can see that when τ varies in the range of 1.45 to 1.583, the patterns gradually transition to the fragmentation type, and finally to the chaotic type. During the process of transition, the patterns show multiplication of spatial periods.

Second, we discuss the effect of cross-diffusion with respect to pattern formation. Setting $d_{12} = 0.01$ and $d_{21} = 0.5$, the rest of the parameters are the same as in 7, and the corresponding diagrams of different τ are shown in Fig. 8. By making a comparison between Fig. 8 and Fig. 7, we find that the patterns in the two sets of diagrams are similar, e.g., the patterns in Fig. 7(a) and Fig. 8(a) are both in the homogeneous stable state. In contrast, the patterns of Fig. 7(f) and Fig. 8(f) are both caused by chaotic attractors and split to pieces. In addition, we observe that the size of the patterns in Fig. 8 is not the same as that in Fig. 7, e.g., Fig. 7(b) and Fig. 8(b). Cross-diffusion seems to have an effect on the size of the patterns. In addition, increasing the cross-diffusion coefficient, in particular, set $d_{12} = 0.02$ and $d_{21} = 0.8$, the patterns of different τ are shown in Fig. 9. For comparison of Fig. 8 and Fig. 9, we see that Fig. 9 has richer patterns than Fig. 8. Cross-diffusion seems to have an effect on both the size and type of patterns formation.

Figure 8

On the $200\times 200$ lattices, $D_{1} = 0.03$, $D_{2} = 1$, $d_{12} = 0.01$, $d_{21} = 0.5$, and $\delta = 50$, and the spatial patterns caused under flip-Turing instability and chaos for an iteration step size of 20000

Figure 9

On the $200\times 200$ lattices, $D_{1} = 0.03$, $D_{2} = 1$, $d_{12} = 0.02$, $d_{21} = 0.8$, and $\delta = 50$, and the spatial patterns caused under flip-Turing instability and chaos for an iteration step size of 20000

4.2 Neimark–Sacker bifurcation and its corresponding patterns

At first, we explain numerically the Neimark–Sacker bifurcation results in Theorem 4. Thereafter, we unite this as well as the Turing bifurcation and obtain the associated patterns. The system parameters are fixed to $r=1$, $m=0.5$, $K=5$, $b=0.01$, $a=0.72$ and the computational grids are $n\times n=200\times 200$.

We can directly compute to obtain the fixed point $(u_{3},v_{3})=(3, 1)$. According to Theorem 4, when $\tau =\tau _{0}=1.90476$, map (10) undergoes the Neimark–Sacker bifurcation, whose bifurcation diagram can be found in Fig. 10(a), and the original value is $(u_{3}+0.001,v_{3}+0.001)$. We can see that the Neimark–Sacker bifurcation could result in chaos. To quantify chaotic and nonchaotic behavior, maximum Lyapunov exponent is plotted against Fig. 10(a) (see Fig. 10(b)). From the local magnification in Fig. 10(b), which is shown in Fig. 11, we can be seen that the maximum value of Lyapunov exponent is greater than 0 for τ is about 2.58 and 2.77. This suggests that chaotic behavior may be happening.

Figure 10

(a) Neimark–Sacker bifurcation diagram; (b) maximum Lyapunov exponent corresponding to Fig. 10(a)

Figure 11

Local amplification of Fig. 10(b)

Below, we utilize phase diagrams to describe the changes in dynamic behavior of Neimark–Sacker bifurcation diagram from fixed point to chaotic behavior. If $\tau =1.81$, then $\tau <\tau _{0}$. We compute the eigenvalues of $J(u_{3},v_{3})$ and obtain $\lambda _{1,2}=0.7285\pm \mathrm{i}0.665036$, $|\lambda _{1,2}|=0.9864<1$. Thus, the homogeneous steady state $(u_{3},v_{3})=(3,1)$ of map (10) is a stable focus. Next, corresponding to Neimark–Sacker bifurcation diagrams as shown in Fig. 10(a), our next step is to use phase diagrams to show the dynamical transmission from stable focus to chaotic behavior by increasing τ, as shown in Fig. 12. When $\tau =1.81$, it is a stable focus, as in Fig. 12(a). When $\tau =2.2$ and $\tau >\tau _{0}$, by Theorem 4 and Figure 10(a), this is an unstable focus and bifurcates into a circle of attraction-invariant, see Fig. 12(b). In addition, letting $\tau =2.41$, 2.45, 2.51, we get a few other invariant circles with bigger amplitudes seen in Figs. 12(c), (d), (f). Keep increasing the value of τ, e.g., $\tau =2.49$ and $\tau =2.62$, and we obtain the period window that corresponds to Fig. 10(a) as shown in Fig. 12(e), (h), respectively. Based on the maximum Lyapunov exponent in Fig. 11 such that $\tau =2.58$ and $\tau =2.77$, we obtain two different chaotic attractors, see Fig. 12(g), (i). As τ increases, Fig. 12 shows the dynamical transition from the focus to the invariant circle, through the periodic windows in between, and finally to the chaotic attractor.

Figure 12

Phase diagrams corresponding to different τ values of Fig. 10, representing dynamic transitions of map (10)

The relationship between $Z_{m}$ and τ is plotted in Fig. 13(a) for the threshold value for the Turing instability, i.e., $\tau ^{\prime }\approx 1.9047066$. Next, we plot the Neimark–Sacker and Turing bifurcation curves, see Fig. 13(b). Pure Turing and Neimark–Sacker–Turing instability regions are obtained. Both regions on the right do not form a pattern, which are a pure Neimark–Sacker bifurcation region and a homogeneous steady state region.

Figure 13

(a) $Z_{m}-\tau$ diagram of the Turing bifurcation; (b) $\tau -D_{2}$ diagram shows regions of patterns formation

Below, we show the process of forming spatial patterns. The initial state is the homogeneous steady state $(u_{3},v_{3})$ with random perturbations.

First, we show the patterns caused by pure Turing, Neimark–Sacker–Turing, and chaos with self-diffusion. Let $D_{1}=0.06$, $D_{2}=0.1$, $d_{12} = 0$, $d_{21} = 0$, $\delta = 10$, and its spatio-temporal patterns can be seen in Fig. 14. When $\tau =1.81$, which corresponds to Fig. 12(a), $\tau <\tau _{0}$ and $\tau <\tau ^{\prime }$. Neimark–Sacker and Turing bifurcations do not occur at this time. From Fig. 14(a), we can see that the biomass v of immune cells is homogeneously distributed at a homogeneous level. At $\tau =1.90471$, $\tau ^{\prime }<\tau <\tau _{0}$, pure Turing instability is present in CMLs models. We can observe the spotted patterns from Fig. 14(b). Assuming $\tau =2.2$, so $\tau >\tau ^{\prime }$, $\tau >\tau _{0}$, Neimark–Sacker–Turing instability is present in CMLs models. The spotted patterns are distorted as can be seen in Fig. 14(c). Continuing to change the τ, we observe different kinds of patterns, e.g., spots and spirals, see Figs. 14(d)–(h). When τ reaches about 2.62, the patterns break up and we get mosaics patterns. As τ increases, the patterns eventually take on a completely disordered and chaotic appearance, as shown in Figs. 14(i)–(j). During the change of patterns, we see that the patterns shift from speckles to stripes, to spirals, to mosaics with increasing irregularity. It is worth noting that at the period windows, the spatial pattern is spiral when $\tau =2.49$ and mosaic when $\tau =2.62$. And for two different chaotic attractors, the spatial pattern is of mosaic type, but as the value of τ becomes larger, the spatial pattern becomes more scattered. In addition, we find that there are periods in these patterns during the simulation. These periods might be related to the periods of the fixed-point invariant cyclic bifurcation caused by the Neimark–Sacker bifurcation.

Figure 14

On the $200\times 200$ lattices, $D_{1}=0.06$, $D_{2}=0.1$, $d_{12}=0$, $d_{21}=0$, and $\delta = 10$, and the spatial patterns caused by pure Turing instability, Neimark–Sacker–Turing instability and chaos for 20000 iterations

Then, we discuss cross-diffusion effects on pattern formation. Suppose $d_{12} = 0.01$, $d_{21} = 0.02$, with other parameters the same as in Fig. 14. The corresponding patterns for different τ can be found in Fig. 15. Comparing Fig. 14 with Fig. 15, it can be noticed that the kinds of Turing patterns are similar in the two sets of diagrams, but the sizes of the patterns are very different. In addition, by increasing the cross-diffusion coefficient, in particular, so that $d_{12} = 0.02$ and $d_{21} = 0.08$, there are different τ patterns as shown in Fig. 16. We find that the latter has a richer pattern when comparing Fig. 15 with Fig. 16. Cross-diffusion seems to have an effect on both the size and type of patterns formation.

Figure 15

On the $200\times 200$ lattices, $D_{1}=0.06$, $D_{2}=0.1$, $d_{12}=0.01$, $d_{21}=0.02$, and $\delta = 20$, the spatial patterns caused by pure Turing instability, Neimark–Sacker–Turing instability and chaos for 20000 iterations

Figure 16

On the $200\times 200$ lattices, $D_{1} = 0.06$, $D_{2} = 0.1$, $d_{12} = 0.02$, $d_{21} = 0.08$, and $\delta = 20$, and the spatial patterns caused by pure Turing instability, Neimark–Sacker–Turing instability and chaos for 20000 iterations

5 Conclusions

This paper focuses on the homogeneous and heterogeneous dynamics of spatio-temporal discrete cancer growth model with strong Allee effects for self-diffusion and cross-diffusion. The continuous form of cancer growth model has attracted considerable attention, but still little attention has been focused on the discrete form. The discrete form results in more complicated nonlinear behavior and new dynamical phenomena compared to the current continuous form work. In simulations we observe many new and interesting patterns such as spirals, mosaics, spots, and chaotic patterns. There are reasons why we chose to discretize the continuous time model: Firstly, continuous time models are often unable to find an exact analytical solution, so we need to resort to the discretization method. Secondly, the data about tumor cells spreading are inherently discrete. Furthermore, the dynamics in the lattice are more accurately represented in a fragmented environment of tumor cells spreading and reflect the complexity of pattern formation more effectively than in a continuous space.

First, we analyze the dynamical behavior of a class of strong Allee effect cancer growth model with self-diffusion and cross-diffusion through CMLs models, totally different from previous studies in literature. Second, we obtain the conditions for the existence and stability with nonnegative fixed points and the threshold parameter conditions where bifurcation occurs, respectively, via theoretical analyses. Lastly, numerical simulations clarify the theoretical results, and we show that CMLs models can perform complicated dynamical behaviors with periodic orbits, invariant circles, and chaos.

It is worth noting that for homogeneous behavior, CMLs models of cancer growth shows flip bifurcation and chaos, while spatio-temporal discrete models of cancer growth with self-diffusion have Turing instability, which are not possible in the relevant continuous system. As a result, the discrete form of the cancer growth model is more richly characterized by dynamics, in particular the chaotic behaviors. We use Turing and Neimark–Sacker (or flip) bifurcation curves to partition the parameter space of the cancer growth model ($D_{2}$, τ) into four regions, which are referred to as the homogeneous steady state, the pure flip (or Neimark–Sacker), the pure Turing and the flip-Turing (or Neimark–Sacker–Turing) instability regions. For the heterogeneous behavior, there are four mechanisms by which Turing patterns can be produced: pure Turing, flip-Turing, Neimark–Sacker–Turing instability, and chaos. Generally, Turing patterns that are due to flip-Turing instability exhibit spatial doubling of segments. Pure Turing instability induces speckle patterns as fixed point is a focus. In addition, a spotted pattern exhibits periodic changes in time. The pattern is characterized by spirals when the CMLs model shows Neimark–Sacker–Turing instability. During spatial pattern transformations, it is observed that changes in τ affect the size of the patterns, as shown in Figs. 14(b)–(c). Chaos induced patterns show disordered behavior in both time and space, and as τ gets larger, the patterns become more complicated and fragmented.

For discrete forms of cancer growth model, we focus on exploring the formation of patterns in the Neimark–Sacker–Turing and flip-Turing instability regions. In addition, through numerical simulations, we discuss the influence of cross-diffusion on pattern formation and see the effect of cross-diffusion with respect to the size and type of patterns, which show that the patterns are more informal and more complicated. The results of this paper can be used to mathematically understand the process of immune cell and tumor cell interactions. The existence of the flip bifurcation suggests that the number of immune cells can remain stable for a longer period of time when the value of $\tau ^{*}$ is finite, while the number of tumor cells remains constant. The Neimark–Sacker bifurcation reveals the phenomenon that the number of tumor cells and the number of immune cells can co-exist in a stable limiting circle, which implies that the disease may be somehow controlled. The Turing bifurcation is a special case in which the number of tumor cells and immune cells will be stable in a certain steady state when $Z_{m}(\tau )<1$ is satisfied. However, when the condition becomes $Z_{m}(\tau )>1$, this stability will be broken, leading to the instability and the formation of the patterns. The Turing patterns in the discrete model reveal the potential for regular and irregular self-organized structures to coexist in space. This connection between bifurcation, chaos, and pattern formation provides us with the key to a deeper understanding of the dynamic complexity of mathematical models of cancer growth.