1 Introduction

In the environment, the vegetation is widespread. This wealth is in huge danger due to the wide expansion of cities and the over carbon dioxide leak for a reason for our daily activities. The importance of the green cover is to suck the carbon dioxide and release the oxygen using what is known as photosynthesis, which is very important to our survival. For the safety of our planet, it must be a balance between the released carbon dioxide by different living beings and the sucked carbon by the plants by photosynthesis. Global warming and climate change are the direct consequence of the imbalance between releasing and suction of this toxic gas. At the moment, for the subject of reducing the concentration of this gas in the environmental air, the scientists tried to find a replacement of the lost green areas by the overbuilding of new cities. There is some work in augmenting the vegetation cover in different areas in the cities such as in the building (at the roof), gardens, shrubs, which is not enough. The semi-arid regions are the best places to begin. The main issue in these regions is the water, where the availability of this last is low. The quantity of water is crucial in determining the growth and expansion of the vegetation cover. This availability can be affected by many factors, and we mention rainfall, illumination, geographical conditions, graving. Consequently, various vegetation structures appear for different conditions, where this diversity is known by vegetation patterns [2, 3, 10,11,12,13]. In literature, there are numerous models that describe the relationship between water and plants. One of the first models is offered by Klausmerier [13], where the following spatiotemporal model is provided for modeling the connection between plant biomass (denoted by n) and water-limited biomass (denoted by w)

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\partial w}{\partial t}=R-w-wn^2-\upsilon \frac{\partial w}{\partial x}, \\ \frac{\partial n}{\partial t}=wn^2-\delta n + \Delta n. \end{array} \right. \end{aligned}$$
(1.1)

R stands for the precipitation (soil, water) or rainfall rate; \(\delta \) stands for the rate of natural death of plants; \(\upsilon \) is the rate of the slope of terrain (or downhill runoff); the term \(wn^2\) represents the amount of water absorbed by the plant or the long-range rivalry and short-range facilitation, \( \Delta \) is the bidimensional Laplacian operator. It has been confirmed that this model predicts stripe patterns next to the wavelength which agrees with the real-life situation.

It is well known that the plants have the ability to redistribute the water in the soil. This point of view is used for the first time in water-plant interaction by Hardenberg et al. [35], where an expanded model is utilized for describing the feedback impact between vegetation biomass and water. The main idea is to add a cross-diffusion \(\gamma \Delta (w-\beta n)\) in the water equation for modeling this feedback. The obtained system is:

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\partial w}{\partial t}=R-w-wn^2-\upsilon \frac{\partial w}{\partial x} + \gamma \Delta (w-\beta n), \\ \frac{\partial n}{\partial t}=wn^2-\delta n + \Delta n, \end{array} \right. \end{aligned}$$
(1.2)

where \(\gamma \) is the rate corresponding to diffusion of the water in lack of vegetation. Using Darcy’s law [35], we can model the transit of water through the soil. More precisely, the flux f is proportional to the water metric potential \(\varphi \) (which means that \(f a - \nabla . \varphi \)). The alternation of water due to the transit is \(- \nabla . f a \nabla ^2 . \varphi . \) By taking the following suction functional \(\varphi = w -\beta n \) where w is the metric potential for the soil (see also [35]). In fact, the slope of terrain or the downhill runoff flow velocity is particular for some places not others, which means that it is not a necessary condition for having patterns. A numerical investigation is given for predicting wavelength, wave speed. Some works appear for describing the spatial patterns in the presence of cross-diffusion as [11, 35]. Based on the best of our knowledge, the main mathematical achievement of these types of models is the dynamics introduced by Turing patterns. Further, for the subject of revealing the spatiotemporal patterns near Turing bifurcation point, where the amplitude equation restricted at Turing bifurcation point is used. The vegetation patterns was and still the subject of interest of many recent researches we mention as example the researches [1, 5, 6, 18,19,20,21, 27, 33, 34].

Very recently, the intensity of the inner competition reaction is considered in [37]. An extended model to Klausmeier’s model (1.2) is considered. The investigated model is studied in the following form:

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\partial w}{\partial t}=R-w-\frac{wn^2}{1+\tau n^2}-\upsilon \frac{\partial w}{\partial x} , \\ \frac{\partial n}{\partial t}=\frac{wn^2}{1+\tau n^2}-\delta n + \Delta n, \end{array} \right. \end{aligned}$$
(1.3)

where the numerator in the term \(\frac{wn^2}{1+\tau n^2}\) is the water absorbed by a plant as it has been highlighted in the above model (1.1), and the dominator as inner (or local) competition between biomass. The main idea in considering this term is due to the gathering of water by plants using preferential leaking in the vegetated zone and run-on/run-off from desert regions toward grassed ones. The intensity of competition increases for the densely vegetated areas. The issue of the previous competition functional is the unboundedness of this last for the densely vegetated area which is not true mostly for the semi-deserts areas, where the available water for each unit of the biomass of plants may reduce if the inner rivalry induced negative feedback is more intense, then the leaking induces positive feedback. To mention that in [37], it has been proved that the system (1.3) cannot undergo Turing patterns. Indeed, we consider in this paper the presence of the cross-diffusion in the system (1.3) subjects to the homogeneous no-flux boundary conditions, which means that

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\partial w}{\partial t}=R-w-\frac{wn^2}{1+\tau n^2}-\upsilon \frac{\partial w}{\partial X} + \gamma \Delta (w-\beta n),\texttt { } X\in \Omega , t>0, \\ \frac{\partial n}{\partial t}=\frac{wn^2}{1+\tau n^2}-\delta n + \Delta n,\texttt { } X\in \Omega , t>0, \\ \frac{\partial w}{\partial \mathbf{v }}=\frac{\partial n}{\partial \mathbf{v }}=0,\texttt { } X\in \Omega , t>0, \\ w(X,0)=w_0(X)\ge 0, n(X,0)=n_0(X)\ge 0 ~~~X\in \Omega , \end{array} \right. \end{aligned}$$
(1.4)

where \(\Omega \) is assumed to be a bounded set of \({\mathbb {R}}^2\) with a smooth boundary. The importance of considering such a model is for generalizing the previous models where the model (1.4) considers various factors such as feedback regulation, water redistribution, the intensity of inner rivalry where each one of the previous models lacks these components (partially or entirely). More precisely, the system (1.2) can be considered as a special case of the system (1.4), wherein the absence of cross-diffusion (means that \(\gamma =0\)) the system becomes the system (1.3). Moreover, in the absence of the inner competition of plants, the system returns to the model (1.2). For the nonexistence on both the inner rivalry of plants and the feedback regulation, the system (1.4) becomes system (1.1), which shows the richness of this system. Further, in the absence of slope of downhill (means that \(\upsilon =0\)) the system can imply many works. For further discussions we rewrite the system (1.3) in the absence of the slope of terrain in the following form:

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\partial w}{\partial t}=R-w-\frac{wn^2}{1+\tau n^2} + \gamma \Delta (w-\beta n),\texttt { } X=(x,y)\in \Omega , t>0, \\ \frac{\partial n}{\partial t}=\frac{wn^2}{1+\tau n^2}-\delta n + \Delta n,\texttt { } X=(x,y)\in \Omega , t>0, \\ \frac{\partial w}{\partial \mathbf{v }}=\frac{\partial n}{\partial \mathbf{v }}=0,\texttt { } X=(x,y)\in \Omega , t>0, \\ w(x,y,0)=w_0(x,y)\ge 0, n(x,y,0)=n_0(x,y)\ge 0. \end{array} \right. \end{aligned}$$
(1.5)

As it has been previously mentioned that the system (1.5) implies many recent works, such as the case of the absence of inner rivalry \(\tau =0\) is discussed in [11], where the existence of Turing patterns, and for discussing the spatiotemporal patterns near the Turing bifurcation points the Amplitude equation is used. Ref. [36] discuses the pattern formations in detail.

Indeed, it is proved in [37] that the system (1.3) has no Turing bifurcation. Consequently, the main focus was on studying the effect of the impact of the slop of terrain and the internal competition between the plants on the vegetation patterns. Here, we will focus on studying the impact of cross-diffusion in the water equation which is generated by taking into consideration the redistribution of the water in the soil next to the internal competition between the plants on the vegetation spatial patterns. Recently, it has been confirmed by [11] that this cross-diffusion generates Turing patterns for the model (1.2) (which is a particular case of our model \(\tau =0\)). Turing patterns have the responsibility for determining the complex pattern that we can get in the real world. On the other hand, The Hopf bifurcation is also very important for determining the existence of the periodic solutions for the spatial systems where the presence of periodic orbits can explain the seasonality of the vegetation grows, which agrees with the real-world situation. To mention that the spatial Hopf bifurcation has never been proved for the vegetation model, and no one can deny its importance in predicting the vegetation patterns. Further, the present paper is not restricted to proving these kinds of patterns. Our main objective is to determine the patterns generated by the intermingling of Turing and Hopf bifurcations, which known as Turing-Hopf bifurcation. To highlight that Turing-Hopf bifurcation is recently investigated which elaborated by the intermingling of both Hopf and Turing bifurcation for different values of wavenumber, we refer as an example the papers [4, 9, 14, 29,30,31]. Turing–Hopf bifurcation has never been studied for vegetation models. Indeed, it has been ensured by many researchers that this kind of bifurcation generates very complex dynamics. To the best of our knowledge the existence of Turing–Hopf bifurcation is confined in one-dimensional space. Thus, our second perspective is to prove the existence of this kind of co-dimensional bifurcation in the case of two-dimensional spatial variables. In literature, for analyzing the spatiotemporal behavior near the Turing-Hopf bifurcation, the normal form on the center of the manifold [31] is always considered. In this paper, by using the Amplitude equations restricted at the Turing-Hopf bifurcation point, is used. Also to mention there are a few works that deal with the effect of cross-diffusion (positive constants of cross-diffusion) on the value of Turing-Hopf bifurcation. But in this paper, we consider another kind of cross-diffusions (negative cross-diffusion \(\gamma \beta \Delta n\)).The Turing-Hopf bifurcation can generate important patterns that consist of a different distribution of the water and the vegetation cover which agrees with the real-life situation. Periodic solutions can model the seasonality of vegetation growth. Studying the vegetation model (1.5) can help the scientists to determine the right condition for the evolution of the vegetation cover mostly in desert and semi-desert areas. Besides, Determining global behavior as example [38,39,40,41,42,43,44,45,46] and local behavior [7, 8, 22,23,24,25,26, 28] is very important to anticipate the biological evolution of the species the the degree of the spread of infectious diseases, where it provides some the time and appropriate (or optimal) suggestion for having the best result (as avoiding the extinction of species, and guaranteeing the stop of the spread of some infectious diseases for epidemiological perspectives). For the ecological and the mathematical perspective, we organize the paper in the following manner:

  • In the next section, the well-posedness of the problem is proved using the theory of invariant regions.

  • Section three is the main part of the paper, where some results are proved for the first time for the vegetation models. In fact, the dynamics introduced in the absence of diffusion where the existence of Hopf bifurcation is proved (actually, the authors in [37] do not deal with a possible bifurcation that the system can undergo) this result is expanded in the presence of diffusion where the rainfall rate is used as a bifurcation parameter. The existence of Turing bifurcation is proved under some appropriate conditions on the model’s parameters. Further, the existence of Turing-Hopf bifurcation is also investigated next to its effect on the spatiotemporal patterns of the solution where the amplitude equations are used to determine these complex patterns.

  • A numerical simulation section is offered to emphasize the obtained theoretical results and shows the complex dynamics generated by the presence of Turing-Hopf bifurcation.

  • A final discussion and remarks conclude the paper.

2 Existence, positivity of solution

In this section, we shall prove the existence of positive solution for the system (1.5), where the result is resumed in the following theorem:

Theorem 2.1

Assume that \(w_0(x,y)\ge 0\) and \(n_0(x,y)\ge 0\) then the system (1.5) has a unique positive solution.

Proof

By putting

$$\begin{aligned} f_1(w,n)=R-w-\frac{wn^2}{1+\tau n^2}, \text { and }f_2(w,n)=\frac{wn^2}{1+\tau n^2}-\delta n, \end{aligned}$$

and the linear operator

$$\begin{aligned} \Lambda =\left( \begin{array}{cc} \Delta &{}\quad 0 \\ 0 &{}\quad \Delta \end{array} \right) , \end{aligned}$$

The system (1.5) can be rewritten in the following structure

$$\begin{aligned} \frac{dU}{dt}=D\Lambda U+ F(U), \end{aligned}$$
(2.1)

where

$$\begin{aligned} D=\left( \begin{array}{cc} \gamma &{}\quad -\gamma \beta \\ 0 &{}\quad 1 \end{array} \right) ,\quad \text { and }\quad F=\left( \begin{array}{l} f_1 \\ f_2 \end{array} \right) , \end{aligned}$$

It remains to determine the space at which the linear operator \(\Lambda \) generates a \(C_0\) semi–group, the appropriate space is \(\left( H^1({\mathbb {R}}^2) \right) ^2\) (which means that \( H^1({\mathbb {R}}^2) \times H^1({\mathbb {R}}^2) \)). Under the norm \(\Vert U \Vert =\left( \int _{{\mathbb {R}}^2} \left( |w|^2+|\nabla w|^2+|n|^2+|\nabla n|^2 \right) dX \right) ^{\frac{1}{2}},\) where \(X=(x,y)^T \in {\mathbb {R}}^2\) the completion of the space \(\left( C_0 ^\infty ({\mathbb {R}}^2)\right) ^2\) is \(\left( H^1({\mathbb {R}}^2) \right) ^2\) where the linear operation is well defined in this space.

Defining \(\Lambda U = (\triangle w, \triangle n )\) where \(U=(w,n) \in D(\Lambda ):= \left( H^1({\mathbb {R}}^2) \right) ^2\), which means that \(\Lambda \) generates a \(C_0\) semigroup. Now let us focus on verifying that \(f_i, i=1,2\) are a Lipschitz functionals. By putting

$$\begin{aligned} |f_1 (w_1,n_1)-f_1 (w_2,n_2) | =\left| -w_1-\frac{w_1n_1 ^2}{1+\tau n_1 ^2}+w_2+\frac{w_2 n_2 ^2}{1+\tau n_2 ^2}\right| , \end{aligned}$$

we get

$$\begin{aligned} \begin{array}{l} \le |w_1-w_2| +\left| \frac{w_1n_1 ^2}{1+\tau n_1 ^2}-\frac{w_2 n_2 ^2}{1+\tau n_2 ^2}\right| , \\ \le |w_1-w_2| +\frac{M^2}{1+\tau M^2}(|w_1-w_2|+|n_1-n_2|),\\ \le \frac{1+(1+\tau )M^2}{1+\tau M^2}|w_1-w_2| +\frac{M^2}{1+\tau M^2}|n_1-n_2|, \end{array} \end{aligned}$$

Hence, \(f_1\) is suitably Lipschitz, By the same manner we have

$$\begin{aligned} |f_2 (w_1,n_1)-f_2 (w_2,n_2) | =\left| \frac{w_1n_1 ^2}{1+\tau n_1 ^2}-\delta n_1-\frac{w_2 n_2 ^2}{1+\tau n_2 ^2}+\delta n_2\right| , \end{aligned}$$

which leads to

$$\begin{aligned} \begin{array}{l} \le \delta |n_1-n_2| +\left| \frac{w_1n_1 ^2}{1+\tau n_1 ^2}-\frac{w_2 n_2 ^2}{1+\tau n_2 ^2}\right| , \\ \le |w_1-w_2| +\frac{M^2}{1+\tau M^2}(|w_1-w_2|+|n_1-n_2|),\\ \le \frac{1+(1+\tau )M^2}{1+\tau M^2}|n_1-n_2| +\frac{M^2}{1+\tau M^2}|w_1-w_2|, \end{array} \end{aligned}$$

which means that \(f_2\) is also suitably Lipschitz. It is not though to check that

$$\begin{aligned}&|F(w_1,n_1)-F(w_2,n_2)|\le \left( \frac{1+(1+\tau )M^2}{1+\tau M^2}+ \frac{M^2}{1+\tau M^2}\right) |w_1-w_2|\\&\quad +\left( \frac{1+(1+\tau )M^2}{1+\tau M^2}+ \frac{M^2}{1+\tau M^2}\right) |n_1-n_2| \end{aligned}$$

thus

$$\begin{aligned} |F(w_1,n_1)-F(w_2,n_2)|\le \frac{1+(2+\tau )M^2}{1+\tau M^2} \left( |w_1-w_2| + |n_1-n_2| \right) . \end{aligned}$$

Using the standard theory of \(C_0\) semi group (see [17]) we deduce the existence and the uniqueness of a positive solution. Here we considered the positivity and the boundedness of solution where a common bound denoted by M is used for simplifying the computations. The proof is completed. \(\square \)

3 Complex spatiotemporal patterns generated by taking into consideration of the plant’s effect on the redistribution of the water in the soil

3.1 Linear analysis: Hopf, Turing, Turing-Hopf bifurcations

This section demonstrates the existence of the Turing-Hopf bifurcation. First of all, the system (1.5) has the following homogeneous steady states

  • \(E_0=(R,0)\) which models the desert state.

  • \(E_\pm =(w_\pm ,n_\pm )\) which models the coexistence of both water and vegetation biomass, where

    $$\begin{aligned} w_\pm = \frac{\delta (1+\tau n_\pm ^2)}{n_\pm } \texttt { and } n_\pm = \frac{1}{2\delta (1+\tau )}\left( R \pm \sqrt{R^2-4(1+\tau )\delta ^2} \right) , \end{aligned}$$

    which has its biological significant if and only if \(R>R_\mathrm{min}:= 2\delta \sqrt{1+\tau }\)

For determining the local stability of the vegetation steady states \(E_\pm \) we use the translation to the origin in the form

$$\begin{aligned} V(t)=\left( \begin{array}{c} {\widetilde{w}} \\ {\widetilde{n}} \end{array}\right) =\left( \begin{array}{c} w-w_\pm \\ n-n_\pm \end{array}\right) , \end{aligned}$$

Neglecting the tilde for simplicity, the system (1.5) can be expressed in the following manner:

$$\begin{aligned} V_t=J_\pm \left( \begin{array}{c} w \\ n \end{array}\right) + \left( \begin{array}{c} f_1(w,n) \\ f_2(w,n) \end{array}\right) +D\triangle V, \end{aligned}$$
(3.1)

where Vt is the partial derivative of V with respect to t and \(J_\pm \) is the Jacobian matrix of the system (1.5) in the absence of diffusion at the vegetation equilibrium state \(E_\pm \) which is given as follows:

$$\begin{aligned} J_\pm =\left( \begin{array}{cc} -\frac{n_\pm ^2}{1+\tau n_\pm ^2}-1 &{} \frac{-2\delta }{1+\tau n_\pm ^2} \\ \frac{n_\pm ^2}{1+\tau n_\pm ^2} &{} \frac{2\delta }{1+\tau n_\pm ^2}-\delta \end{array} \right) , \end{aligned}$$
(3.2)

we put

$$\begin{aligned} \begin{array}{cc} \alpha _{11}= -\frac{1+(1+\tau ) n_+ ^2}{1+\tau n_+ ^2} &{} \alpha _{12}= \frac{-2\delta }{1+\tau n_+ ^2} \\ \alpha _{21}= \frac{n_+ ^2}{1+\tau n_+ ^2} &{} \alpha _{22}= \frac{\delta -\delta \tau n_+ ^2}{1+\tau n_+ ^2} \end{array} \end{aligned}$$

and

$$\begin{aligned}&f_1(w,n)=\alpha _1 w^2+ \alpha _2 wn+ \alpha _3n^2+ \alpha _4 w^3+\cdots ,\\&f_2(w,n)=\beta _1 w^2+ \beta _2 wn+ \beta _3n^2+ \beta _4 n^3+\cdots , \\&\begin{array}{ll} \alpha _1= 0 &{} \beta _1= 0\\ \alpha _2= \frac{-2n_+}{(1+\tau n_+ ^2)^2}&{} \beta _2= \frac{2n_+}{(1+\tau n_+ ^2)^2},\\ \alpha _3= \frac{-4n_+(1-\tau n_+ ^2)}{(1+\tau n_+^2)^2}&{} \beta _3=\frac{2w(1-2\tau n_+ ^2)}{(1+\tau n_+ ^2)^3},\\ \alpha _4= 0&{} \beta _4=0. \end{array} \end{aligned}$$

The linearized system of (3.1) at an arbitrary homogeneous steady state is expressed as follows:

$$\begin{aligned} V_t=J_\pm \left( \begin{array}{c} w \\ n \end{array}\right) +D \left( \begin{array}{c} \triangle w \\ \triangle n \end{array}\right) . \end{aligned}$$
(3.3)

The linear stability of the vegetation homogeneous steady states \(E_\pm \) implies that the system (3.3) admits the following solution

$$\begin{aligned} \left( \begin{array}{c} w \\ n \end{array}\right) = \left( \begin{array}{c} c_1 \\ c_2 \end{array}\right) exp(\lambda t +i \upsilon \cdot x). \end{aligned}$$

The characteristic equation associated with the above equation (3.3) is given as:

$$\begin{aligned} \Psi _\upsilon \triangleq \lambda ^2 -T_\upsilon \lambda + D_\upsilon =0,\text { }\upsilon \in {\mathbb {N}}_0, \end{aligned}$$
(3.4)

where

$$\begin{aligned} \begin{array}{l} T_\upsilon =\frac{\delta -1 - (1+\tau +\delta \tau )n_\pm ^2}{1+\tau n_\pm ^2}-(\gamma +1) \upsilon ^2, \\ D_\upsilon = \gamma \upsilon ^4 +\upsilon ^2 \left( \frac{-(1+(1+\tau )n_\pm ^2)+\gamma (\delta + (\beta -\tau )n_\pm ^2)}{1+\tau n_\pm ^2} \right) +\frac{\delta (1+\tau )\left( n_\pm ^2 -\frac{1}{1+\tau }\right) }{1+\tau n_\pm ^2}. \end{array} \end{aligned}$$
(3.5)

The roots of the characteristic Eq. (3.4) can be found as:

$$\begin{aligned} \lambda _\upsilon =\frac{1}{2}\left( T_\upsilon \pm \sqrt{T_\upsilon ^2- 4 D_\upsilon }\right) , \end{aligned}$$

In order to set the stability conditions generated by the absence of cross-diffusion we consider the following theorem

Theorem 3.1

(Dynamics introduced by the absence of diffusion) Assume that \(R>R_\mathrm{min}\) holds, the system (1.5) in the case of absence of diffusion (means that \(\upsilon =0 \)) verifies the following aspects

  1. (i)

    The equilibrium state \(E_-\) is unconditionally unstable.

  2. (ii)

    For \(\delta <1\) then \(E_+\) is unconditionally stable.

  3. (iii)

    The equilibrium state \(E_+\) is stable for (\(\delta>1 \text { and }R>R_{H,0+}:=\delta (1+\tau ) \sqrt{\frac{\delta -1}{1+\tau +\delta \tau }}\)) and unstable for (\(\delta >1 \text { and }R<R_{H,0+}\))

  4. (iv)

    For \(\upsilon =0 \) (or in the absence of diffusion) next to the condition \(\delta >1\) the system (1.5) undergoes Hopf bifurcation at \(R=R_{H,0+}\).

Proof

Putting \(n=0\) in \(D_n\) defined in (3.5), we obtain:

$$\begin{aligned} D_0=\frac{\delta (1+\tau )\left( n_\pm ^2 -\frac{1}{1+\tau }\right) }{1+\tau n_\pm ^2}, \end{aligned}$$
(3.6)

and by doing some formal calculations, we have:

$$\begin{aligned} n_- -\frac{1}{\sqrt{1+\tau }}=\frac{-2(R-2\delta \sqrt{1+\tau })}{\sqrt{1+\tau } \left( R-2\delta \sqrt{1+\tau }+\sqrt{R^2-4\delta ^2 (1+\tau )} \right) } < 0. \end{aligned}$$

The last result leads to deduce that \(D_0<0\) for \(R>R_\mathrm{min}\), which means that the homogenous steady-state \((w_-,n_-)\) is saddle. Hence, it is unstable. Thus (i) holds. Besides,

$$\begin{aligned} n_+ -\frac{1}{\sqrt{1+\tau }}=\frac{(R-2\delta \sqrt{1+\tau })(R+2\delta \sqrt{1+\tau })+\sqrt{R^2-4\delta ^2 (1+\tau )}}{\left( R+\sqrt{R^2-4\delta ^2 (1+\tau )}+\sqrt{1+\tau }\right) \delta (1+\tau )}>0. \end{aligned}$$

Thus, \(D_0>0\) under the condition \(R>R_\mathrm{min}\), this, takes us to deduce that \(T_0\) determines the stability of the equilibrium \(E_+\) in the absence of cross-diffusion. For \(n=0\) the sign of \(T_0\) determines the local behavior of the equilibrium state \(E_+\). In other words, \(T_0<0\) holds if and only if

$$\begin{aligned} \delta >1 + (1+\tau +\delta \tau )n_+ ^2. \end{aligned}$$
(3.7)

It is easy to verify that for \(\delta <1\) one has \(T_0<0\); which means that \(E_+\) is locally stable under this condition (then (ii) is checked). Now focusing on studying the local behavior offered by the condition \(\delta >1\) and \(R>R_\mathrm{min}\). Under these conditions the inequality (3.7) becomes:

$$\begin{aligned} R>R_{H,0}^+:=\sqrt{\frac{\delta ^2(\delta -1)(1+\tau )^2}{1+\tau +\delta \tau }}+\sqrt{\frac{\delta ^2 (1+\tau +\delta \tau )}{\delta -1}}. \end{aligned}$$
(3.8)

It is easy to see that under the condition \(\delta >1\), \(E_+\) is stable for \(R>R_{H,0}^+\) and unstable for \(R<R_{H,0}^+\). Further, for \(R=R_{H,0}^+\) the characteristic equation has two purely imaginary roots \(\pm \sqrt{\frac{\delta (1+\tau )\left( n_+ ^2 -\frac{1}{1+\delta }\right) }{1+\tau n_+ ^2}}\) which means that for showing the existence of Hopf bifurcation it remains to verify the transversality condition which is given as follows:

$$\begin{aligned} \left. \frac{d \mathfrak {R}{(\lambda )}}{dR}\right| _{R=R_{H,0}^+}=\left. n_+(2-\delta +(1+\tau )(1+\delta )n_+^2)\frac{dn_+}{dR}\right| _{R=R_{H,0}^+}. \end{aligned}$$

By using the fact that \(n_+>\frac{1}{\sqrt{1+\tau }}\) leads to deduce that \(2-\delta +(1+\tau )(1+\delta )n_+ ^2>3>0\). Further

$$\begin{aligned} \left. \frac{dn_+}{dR}\right| _{R=R_{H,0}^+}=\frac{1}{2\delta (1+\tau )}\left( 1+\frac{2R_{H,0}^+}{2\sqrt{\left( R_{H,0}^{+}\right) ^{2} +4\delta ^2(1+\tau )}}\right) >0. \end{aligned}$$

Combining the two results, we deduce that \(\left. \frac{d \mathfrak {R}{(\lambda )}}{dR}\right| _{R=R_{H,0}^+}>0\) which means that the system (1.5) undertakes Hopf bifurcation for \(n=0\) at \(R=R_{H,0}^+\). This puts an end to the proof. \(\square \)

It has been confirmed by Turing [32] that diffusion can lead to instability where for some values of the diffusion parameters, we can have instability, such as instability is called by diffusion-driven instability or by Turing instability. In fact, this kind of instability appears when the following conditions hold true:

$$\begin{aligned} \mathfrak {I}{\lambda _\upsilon }\ne 0,\mathfrak {R}{\lambda _\upsilon }=0, k=k_T\ne 0. \end{aligned}$$

where \(\mathfrak {R}\) and \(\mathfrak {I}\) stand for the real part and the imaginary part functionals, respectively. It is not tough to prove that the following equation is a sufficient condition for having Turing bifurcation:

$$\begin{aligned}&\delta -1-(1+\tau +\delta \tau )n_+ ^2+\sqrt{\frac{|\delta -1 -(1+\tau +\delta \tau )n_+ ^2|}{\gamma (1+\delta n_+ ^2)}}\left[ (1+\tau )n_+ ^2+1+2\gamma \beta \delta +2\gamma \delta \tau n_+ ^2\right] \\&\quad +\delta (1+\tau )\left( n_+^2 -\frac{1}{1+\tau }\right) =0. \end{aligned}$$

For the existence of Hopf bifurcation for the spatiotemporal vegetation model (1.5) we set the following theorem.

Theorem 3.2

(The existence of Hopf bifurcation for the spatial system) Assume that \(R>R_\mathrm{min}\), then the following results arises.

  1. (i)

    The system (1.5) cannot undergo Hopf bifurcation at \(E_-\).

  2. (ii)

    For \(\delta <1\): The system (1.5) cannot undergo Hopf bifurcation at the constant steady state \(E_+\).

  3. (iii)

    For \(\delta >1\): There exists a wave number denoted by \(\vartheta _H \ge 0\) such that system (1.5) undertakes Hopf bifurcation for the constant steady state \(E_+\) at \(R=R_{\vartheta ,H}\), where this homogeneous steady state is locally stable for \(R>\max \{R_{0,H},R_\mathrm{min}\}\) and unstable for \(R_\mathrm{min}<R<R_{0,H}\). Further, a spatially homogeneous periodic solution appears for \(\vartheta =0\) and nonhomogeneous ones appear for \(\vartheta =1,2,\ldots ,\vartheta _H\).

Proof

It has been proved in Theorem 2.1 that \(D_0 <0\) for the equilibrium \(E_-\) which leads to deduce that there exists a positive integer (wavenumber) verifying \(D_\vartheta <0\) (\(\vartheta =0\)) which leads to claim that this homogeneous steady state stays unstable in the presence of the diffusion which confirms the obtained result in (i).

Passing to the second part of the proof. It is easy to see that the condition \(T_\vartheta =0\) is an indispensable but not enough condition for getting Hopf bifurcation. Obviously, by taking a look at (3.5) we can deduce that under \(\delta <1\) we get \(T_\vartheta <0\), which leads us to deduce that the system (1.5) cannot possess Hof bifurcation at the equilibrium \(E_+\), which emphasizes (ii).

Now, it remains the main part of the proof, where we will proceed to show the existence of Hopf bifurcation. In addition to the main assumption \(R>R_\mathrm{min}\), we suppose that \(\delta >1\). In what follows, we choose the rainfall constant as a bifurcation parameter. It is well known that Hopf bifurcation occurs if the characteristic equation (3.4) has purely imaginary roots in addition to the transversality condition, which equivalent to

$$\begin{aligned} T_\vartheta =0\text {, }D_\vartheta \ne 0 \text {, }\quad \left. \frac{d\mathfrak {R}(\lambda )}{dR}\right| _{R=R_{\vartheta ,H}}\ne 0. \end{aligned}$$

\(T_\vartheta =0\) is equivalent to

$$\begin{aligned} n_+ ^2=\frac{\delta -1 -(\gamma +1 )\vartheta ^2}{1+\tau +\delta \tau +(\gamma +1)\tau \vartheta ^2 }, \end{aligned}$$
(3.9)

using the fact that \(\delta >1\) we can make sure that the critical wave numbers such that the system (1.5) could possess Hopf bifurcation are \(\vartheta =0,1,...,\vartheta _c\) where

$$\begin{aligned} \vartheta _c:=\max {\{\vartheta \in {\mathbb {N}}, \delta -1 -(\gamma +1 )\vartheta ^2>0\}}:=\left[ \sqrt{\frac{\delta -1}{\gamma +1}}\right] , \end{aligned}$$

where [.] represents the integer part function. Now, considering the wave number be part of the set \(\{0,1,...,\vartheta _c\}\) and making use of the explicit formula of \(n_+\), (3.9) becomes

$$\begin{aligned} R+\sqrt{R^2-4\delta ^2(\tau +1)}=2\delta (1+\tau )\sqrt{\frac{\delta -1-(\gamma +1)\vartheta ^2}{1+\tau +\tau \delta +\tau \vartheta ^2 (\gamma +1)}}, \end{aligned}$$

This equivalent to

$$\begin{aligned}&R=R_{\vartheta ,H}:=\delta \frac{(\delta -1-(\gamma +1)\vartheta ^2) + 1 +2\delta \tau }{\sqrt{(\delta -1 -(\gamma +1 )\vartheta ^2)(1+\tau +\delta \tau +(\gamma +1)\vartheta ^2)}}>0,\\&\quad \text { for } \vartheta =0,1,...,\vartheta _c, \delta <1. \end{aligned}$$

For having purely imaginary roots, we must have \(D_\vartheta >0\), it is easy to see that \(D_0 >0\) which means that there exist a wave numbers \(\vartheta _{c_2} >0\) such that \(D_\vartheta >0\) for \(\vartheta \in \{0, 1, ..., \vartheta _{c2}\}\). In fact, by considering \(\vartheta _H =\min \{\vartheta _c,\vartheta _{c2}\}\) we can claim that the characteristic equation (3.4) has a purely imaginary roots \(\pm \sqrt{D_\vartheta }\) where \(\vartheta =\{0,1,...,\vartheta _H\}\). Now it remains to confirm the transversality condition which is given in the following form

$$\begin{aligned} \left. \frac{d\mathfrak {R}}{dR}\right| _{R=R_{H,\vartheta }}=\frac{n_{+,H}(2-\delta +(1+\tau )(1+\delta )n_{+,H}^2)}{2\delta (1+\tau )}\left( 1+ \frac{2R_{H,\vartheta }^+}{2\sqrt{\left( R_{H,\vartheta }^{+}\right) ^{2} +4\delta ^2(1+\tau )}}\right) >0, \end{aligned}$$

where \(n_{+,H}=\frac{R_{H,\vartheta }+\sqrt{R_{H,\vartheta } ^2 -4\delta ^2 (\tau +1)}}{2\delta (1+\tau )}\), \(\vartheta = 0,1,...,\vartheta _H\). The proof is successfully established. \(\square \)

Now let us focus on studying the existence of Turing-Hopf bifurcation. From many papers such as [4, 9, 14, 29,30,31] we can confirm that Turing-Hopf bifurcation appears at a reaction-diffusion system if there exist two different wavenumbers denoted by \(\vartheta _H\) and \(\vartheta _T \ne 0\) where the vegetation system (1.5) undergoes Hopf bifurcation at \(\vartheta =\vartheta _H\) and undergoes Turing bifurcation at \(\vartheta =\vartheta _T\), and the transversality condition holds. It has been proved widely that this kind of bifurcation affects the spatiotemporal behavior of the solution which is going to anticipate this kind of complex dynamics. Indeed, by taking into consideration the results presented in Theorem 3.1, we can conclude that the system (1.5) undertakes Hopf bifurcation at \(R=R_{0,H}\) for the equilibrium state \(E_+\), we denote by \(L_H\) for Hopf bifurcation curve on \(R-\gamma \) plan. In other words, the system (1.5) has Hopf bifurcation for \(\vartheta =\vartheta _H=0\) for the vegetation homogeneous steady state \(E_+\). Now it remains to show the existence of a strictly positive wave number denoted by \(\vartheta _T\) such that the system undergoes Turing bifurcation at different bifurcation parameters than the rainfall rate. In fact, we will choose the diffusion rate of the water biomass \(\gamma \) as a bifurcation parameter. Also, we consider the \(R-\gamma \) plan. Solving \(D_\vartheta =0\) in \(\gamma \) we get:

$$\begin{aligned} \gamma _T (\vartheta )= \frac{\delta (1+\tau )\left( n_+ ^2-\frac{1}{1+\tau }\right) +((1+\tau )n_+ ^2 +1)\vartheta ^2}{\vartheta ^2 \left( -(1+\tau n_+ ^2)\vartheta ^2 +\delta + n_+ ^2 (\beta -\delta \tau ) \right) }, \end{aligned}$$
(3.10)

where \(n_+\) restricted at \(R_{H,0}\) which means that

$$\begin{aligned} \left. n_+\right| _{R=R_{H,0}}:= \frac{1}{2\delta (1+\tau )}\left( R_{H,0} + \sqrt{R_{H,0}^2-4(1+\tau )\delta ^2} \right) , \end{aligned}$$

and \(\vartheta \in \{1,2, ..., \vartheta ^*\}\) where

$$\begin{aligned} \vartheta ^* :=\left[ \sqrt{\frac{\delta +n_+ ^2 (\beta -\delta \tau )}{1+\tau n_+ ^2}}\right] \end{aligned}$$

and \(\beta >\beta _\mathrm{min}:=\max \left\{ 0,\frac{\delta (1-\tau n_+ ^2)}{n_+ ^2}\right\} \). Setting the positive real x where \(1\leqslant x\leqslant \vartheta ^*\). The differentiation of the functional \(\gamma _T (x)\) with respect to x is given as

$$\begin{aligned} \gamma _T ' (x) =\frac{(1+(1+\tau )n_+ ^2 )x^2+2\delta ((1+\tau )n_+ ^2-1)x-\frac{\delta ((1+\tau )n_+ ^2-1)(\delta +n_+ ^2 (\beta -\delta \tau )) }{1+\tau n_+ ^2}}{x^2 \left( -(1+ \tau n_+ ^2)x +\frac{\delta + n_+ ^2 (\beta -\delta \tau )}{1+\tau n_+ ^2} \right) ^2}. \end{aligned}$$

In fact, the sign of \(\gamma ' (x)\) is given as follows

$$\begin{aligned} \gamma _T ' (x)=\left\{ \begin{array}{l}>0 \quad \text { for } x>x^* ,\\<0 \quad \text { for } x<x^*, \end{array} \right. \end{aligned}$$

where

$$\begin{aligned} x^*=\frac{1}{1+(1+\tau ) n_+ ^2}\left\{ \delta ((1+\tau )n_+ ^2-1)+\sqrt{\frac{\delta ((1+\tau )n_+ ^2-1)n_+ ^2}{1+\tau n_+ ^2}(2\delta +1+(1+\tau )n_+ ^2)} \right\} . \end{aligned}$$

This result means that the functional \(\gamma \) reaches its maximum either at 1 or \(\vartheta ^{*^2} \) which allows us to set the following integer:

$$\begin{aligned} \vartheta _T=\left\{ \begin{array}{l} 1\quad \text { if } \gamma _T (1)>\gamma _T (\vartheta ^{*^2}) ,\\ \vartheta ^* \quad \text { if } \gamma _T (1)<\gamma _T (\vartheta ^{*^2}). \end{array} \right. \end{aligned}$$

We proved that there exists a positive intersection point between the Hopf bifurcation curve defined by \(R=R_{0,H}\) in \(R-\gamma \) plan (curve \(L_H\) in Fig. 4) and the Turing bifurcation curve defined by \(\gamma =\gamma _T(\vartheta ^2)\) (\(L_T\) in in Fig. 4). Then we have

$$\begin{aligned} \left. \frac{d\mathfrak {R}\lambda (R)}{dR}\right| _{R=R_{H,0}}=\frac{n_{+,H}(2-\delta +(1+\tau )(1+\delta )n_{+,H}^2)}{2\delta (1+\tau )}\left( 1+\frac{2R_{H,0}}{2\sqrt{\left( R_{H,0}\right) ^{2} +4\delta ^2(1+\tau )}}\right) >0. \end{aligned}$$

where \(n_{+,H}=\frac{R_{H,0}+\sqrt{R_{H,0} ^2 -4\delta ^2 (\tau +1)}}{2\delta (1+\tau )}\), and also we have

$$\begin{aligned} \left. \frac{d\mathfrak {R}\lambda (R)}{dR}\right| _{L_T}=\frac{2n_+}{(1+\tau n_+)^2T_\vartheta }\left[ \vartheta ^2(-1+\gamma (\beta +\tau -\delta \tau ))+\delta (1+2\tau )\right] \left. \frac{dn_+}{dR}\right| _{L_T}\ne 0. \end{aligned}$$

The obtained results are summarized in the following theorem:

Theorem 3.3

Assume that \(R>R_\mathrm{min}\) an \(\delta >1\) then we have:

The Hopf bifurcation curve \(L_H\) (corresponding the wavenumber 0) intersects the Turing bifurcation curve \(L_T\) and a codimension-2 Turing–Hopf bifurcation occurs at the intersect point \((R_{0,H},\gamma _T(\vartheta _T))\), where

$$\begin{aligned} \gamma _T (\vartheta _T)= \frac{\delta (1+\tau )\left( n_+ ^2-\frac{1}{1+\tau }\right) +((1+\tau )n_+ ^2 +1)\vartheta _T ^2}{\vartheta _T ^2 \left( -(1+\tau n_+ ^2)\vartheta _T ^2 +\delta + n_+ ^2 (\beta -\delta \tau ) \right) }, \end{aligned}$$

and

$$\begin{aligned} \vartheta _T=\left\{ \begin{array}{l} 1 \text { if } \gamma _T (1)>\gamma _T (\vartheta ^{*^2}) ,\\ \vartheta ^* \text { if } \gamma _T (1)<\gamma _T (\vartheta ^{*^2}). \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} R>R_{H,0+}:=\delta (1+\tau ) \sqrt{\frac{\delta -1}{1+\tau +\delta \tau }} \end{aligned}$$

Further, for \((R,\gamma )=(R_{0,H},\gamma _T(\vartheta _T))\) the characteristic equation \(\Psi _0\) has a pair of purely imaginary roots \(\pm i\omega _c\) and \(\Psi _{\vartheta _T}\) has a simple zero root, and for (3.4), there are no other roots with zero real part.

3.2 Nonlinear analysis: The spatiotemporal patterns generated by the presence of Turing-Hopf bifurcation

In this subsection, we will derive the amplitude equations restricted at the Turing-Hopf bifurcation point with the help of the weakly nonlinear analysis. The main interest in considering it is to analyze the complex dynamics next to the Turing-Hopf bifurcation. In fact, the Turing mode \(T(\vartheta _T, 0)\) where \(\vartheta _T\) is the wavenumber will interact with the Hopf bifurcation mode (homogeneous) denoted by \(H(0,\omega _H)\) where \(\omega _H\) is the frequency. The solution of the system (1.5) can be expressed in the following manner:

$$\begin{aligned} U=\varepsilon U^{(1)}+\varepsilon ^2 U^{(2)}+\varepsilon ^3 U^{(3)}+\cdots , \end{aligned}$$
(3.11)

where

$$\begin{aligned} U^{(1)}=\Sigma _{\nu } [A_\nu \exp ^{(i\vartheta _j.x)}+B \exp ^{(i\omega _H t)}]+c.c., \end{aligned}$$

where c.c. stands for the conjugate term. By letting \(\kappa =\upsilon ^2 t\) be the slow time and expands w and n. The bifurcation parameters are R and \(\gamma \), then

$$\begin{aligned} \begin{array}{l} \left( {\begin{array}{c}w\\ n\end{array}}\right) =\varepsilon \left( {\begin{array}{c}w_1\\ n_1\end{array}}\right) +\varepsilon ^2 \left( {\begin{array}{c}w_2\\ n_2\end{array}}\right) +\varepsilon ^3 \left( {\begin{array}{c}w_3\\ n_3\end{array}}\right) +\cdots \\ R=R_{H,0}+R^{(2)}\upsilon ^2. \\ \gamma =\gamma _T ^*+\gamma ^{(2)}\upsilon ^2. \end{array} \end{aligned}$$
(3.12)

Suppose that the amplitude \(\Lambda _j\) and the Hopf mode \(\Phi \) are lingeringly changing variable, this means:

$$\begin{aligned} \begin{array}{c} \frac{d\Lambda _j}{dt}=\varepsilon ^2\frac{d\Lambda _j}{d\kappa }+ O(\varepsilon ^2), \\ \frac{d\Phi }{dt}=\varepsilon ^2\frac{d\Phi }{d\kappa }+ O(\varepsilon ^2). \end{array} \end{aligned}$$

Using (3.12) into the system (3.3), and gathering the first and the second the third power of \(\varepsilon \), the sequence of equations can be expressed in the following manner

$$\begin{aligned} \begin{array}{l} O(\varepsilon ):M_\vartheta \left( {\begin{array}{c}w_1\\ n_1\end{array}}\right) =0, \\ O(\varepsilon ^2):M_\vartheta \left( {\begin{array}{c}w_2\\ n_2\end{array}}\right) =-\left( {\begin{array}{c}f^{(1)}_1\\ f^{(1)} _2\end{array}}\right) \\ O(\varepsilon ^3):M_\vartheta \left( {\begin{array}{c}w_3\\ n_3\end{array}}\right) =\frac{\partial }{\partial \kappa }\left( {\begin{array}{c}w_1\\ n_1\end{array}}\right) -M\left( {\begin{array}{c}w_1\\ n_1\end{array}}\right) -\left( {\begin{array}{c}f^{(2)} _1\\ f^{(2)} _2\end{array}}\right) , \end{array} \end{aligned}$$

where the linear system (3.3) can be expressed as follows

$$\begin{aligned}&U_t=\left[ M_\vartheta + \upsilon ^2 M \right] U, \\&M =\left( \begin{array}{ll} \alpha ^{(2)} _{11} + \gamma ^{(2)}\triangle &{} \alpha ^{(2)} _{12}- \gamma ^{(2)}\beta \triangle \\ \alpha ^{(2)} _{21} &{} \alpha ^{(2)} _{22} \\ \end{array} \right) \\&M _\vartheta =\left( \begin{array}{ll} \alpha ^{TH} _{11}-\gamma ^{TH} \vartheta ^2 &{} \alpha ^{TH} _{12} + \gamma ^{TH} \beta \vartheta ^2 \\ \alpha ^{TH} _{21} &{} \alpha ^{TH} _{22} -\vartheta ^2\\ \end{array} \right) \\&\begin{array}{ll} \left( {\begin{array}{c}f^{(1)}_1\\ f^{(1)} _2\end{array}}\right) =\left( {\begin{array}{c}\alpha _1 w^2 + \alpha _2 wn +\alpha _3 n^2\\ \beta _1 w^2 + \beta _2 wn +\beta _3 n^2\end{array}}\right) ,&{} \left( {\begin{array}{c}f^{(2)}_1\\ f^{(2)} _2\end{array}}\right) =\left( {\begin{array}{c}2\alpha _1 w_1 w_2 + \alpha _2 (w_1 n_2 +w_2 n_1) +2\alpha _3 n_1 n_2 +\alpha _4 w_1 ^3\\ 2\beta _1 w_1 w_2 + \beta _2 (w_1 n_2 +w_2 n_1) +2\beta _3 n_1 n_2 +\beta _4 w_1 ^3\end{array}}\right) , \end{array} \end{aligned}$$

and \(\alpha ^{TH} _{ij}=\left. \alpha _{ij}\right| _{R=R_{0,H},\gamma _T ^*}\) \(i,j=1,2\) and \(\gamma ^{TH}=\gamma _T ^*\). Note that

$$\begin{aligned} M _{\vartheta ^*}q=0, ~~ M _{0}q=i\omega _Hp, \end{aligned}$$

which takes us to deduce that

$$\begin{aligned} q=\left( {\begin{array}{c}q_1\\ q_2\end{array}}\right) =\left( {\begin{array}{c}\frac{\alpha _{12}^{TH}+\beta \vartheta ^2}{\vartheta ^2 -\alpha _{11}^{TH}}\\ 1\end{array}}\right) ,~~~~ p=\left( {\begin{array}{c}p_1\\ p_2\end{array}}\right) =\left( {\begin{array}{c}\frac{\alpha _{12}^{TH}+\beta \vartheta ^2}{i\omega _H -\alpha _{11}^{TH}}\\ 1\end{array}}\right) . \end{aligned}$$

Now we set

$$\begin{aligned} \left( {\begin{array}{c}w_1\\ n_1\end{array}}\right) =\sum _{j=1}^3\left( {\begin{array}{c}X_j\\ {\tilde{X}}_j \end{array}}\right) \exp ^{i\vartheta _j x}+\left( {\begin{array}{c}\Phi _1\\ \Phi _2\end{array}}\right) \exp ^{i\omega _H t}+c.c., \end{aligned}$$
(3.13)

where \(|\vartheta _j|^2 =\vartheta _T\), \(j=1,2,3\),  This means that

$$\begin{aligned} \left( {\begin{array}{c}X_j\\ {\tilde{X}}_j \end{array}}\right) =q\Lambda _j,~~~~\left( {\begin{array}{c}\Phi _1\\ \Phi _2\end{array}}\right) =p\Phi . \end{aligned}$$

Based on the above results, we suppose that the type of solutions \(U^{(2)}\) can be expressed in the following manner

$$\begin{aligned} \left( {\begin{array}{c}w_2\\ n_2\end{array}}\right)= & {} \left( {\begin{array}{c}W_0\\ N_0\end{array}}\right) +\sum _{j=1}^m \left[ \left( {\begin{array}{c}W_j\\ N_j\end{array}}\right) \exp ^{i\vartheta _j .x} +\left( {\begin{array}{c}W_{jj}\\ N_{jj}\end{array}}\right) \exp ^{i2\vartheta _j .x} \right] \nonumber \\&+\,\left( {\begin{array}{c}W_H\\ N_H\end{array}}\right) \exp ^{i2\omega _H .t}+\sum _{n\ne j>0}\left( {\begin{array}{c}W_{nj}\\ N_{nj}\end{array}}\right) \exp ^{i(\vartheta _n-\vartheta _j) .x}+\left( {\begin{array}{c}W_{TH}\\ N_{TH}\end{array}}\right) +c.c.\nonumber \\ \end{aligned}$$
(3.14)

By injecting the Eq. (3.14) into \(O(\varepsilon ^2)\) we get

$$\begin{aligned} \begin{array}{l} (\alpha _{11}^{TH}+\gamma \triangle )w_2+(\alpha _{12}^{TH}-\beta \triangle ) n_2=-(\alpha _1 w_1 ^2+\alpha _2 w_1 n_1 + \alpha _3 n_3 ^2), \\ \alpha _{21}^{TH}w_2+(\alpha _{22}^{TH}+\triangle )n_2=-(\beta _1w_1 ^2+\beta _2 w_1 n_1 +\beta _3 n_1 ^2). \end{array} \end{aligned}$$

Then we can put

$$\begin{aligned} \begin{array}{l} W_0=W_{0}^T+W_0 ^H, ~~~ W_0 ^T=\Theta _{1T}N_{0}^T, ~~~ N_0 ^T=\Theta _{2T}|P_j|^2, ~~~W_0 ^H=\Theta _{1H} N_0 ^H,\\ N_0 ^H=\Theta _{2H}|\Phi |^2, ~~~ W_j =q_1 N_j,~~~W_{12}=\Theta _5 N_{12},\\ N_{jj}=\Theta _4 P_j ^2, ~~~N_{12}=\Theta _6 P_1 \overline{P_2}, ~~~ W_H=\Theta _7 N_H, N_H=\Theta _8 \Phi ^2, \\ W_{TH}=\Theta _9 N_{TH},~~~ N_{TH}=\Theta _{10} P_j \Phi , ~~~W_{jj}=\Theta _3 N_{jj}, \end{array} \end{aligned}$$

such that

$$\begin{aligned}&\begin{array}{l} \Theta _{1T}= \frac{\beta _T \alpha _{12}^{TH}\alpha _T \alpha _{22}^{TH}}{\alpha _T \alpha _{21}^{TH}-\beta _T \alpha _{11}^{TH}},\\ \Theta _{1H}= \frac{\beta _H \alpha _{12}^{TH}\alpha _H \alpha _{22}^{TH}}{\alpha _H \alpha _{21}^{TH}-\beta _H \alpha _{11}^{TH}}, ~~~~ \Theta _{2T}= \frac{-\alpha _T}{\alpha _{11}^{TH}\Theta _{1T}+ \alpha _{12}^{TH}},\\ \Theta _{2H}= \frac{-\alpha _H}{\alpha _{11}^{TH}\Theta _{1H}+ \alpha _{12}^{TH}}, ~~~~\Theta _{3}=\frac{\alpha _{2T} \left( \alpha _{22}^{TH}-4\vartheta ^2\right) -\beta _{2T}\alpha _{12}^{TH} +\beta \gamma ^{TH} \vartheta ^2}{\beta _{2T} \left( \alpha _{11}^{TH}-4\gamma ^{TH} \vartheta ^2\right) -\alpha _{2T}\alpha _{21}^{TH}},\\ \Theta _{4}=-\frac{-\alpha _{2T}}{(\alpha _{11}^{TH}-4\vartheta ^2)\Theta _{3}+ \alpha _{12}^{TH}+\beta \gamma ^{TH} \vartheta ^2}, \\ \Theta _{5}=\frac{\alpha _{3T} \left( \alpha _{22}^{TH}-3\vartheta ^2\right) -\beta _{3T}\alpha _{12}^{TH} +\beta \gamma ^{TH} \vartheta ^2}{\beta _{3T} \left( \alpha _{11}^{TH}-3\gamma ^{TH} \vartheta ^2\right) -\alpha _{3T}\alpha _{21}^{TH}}, \\ \Theta _{6}=-\frac{-\alpha _{3T}}{(\alpha _{11}^{TH}-3\vartheta ^2)\Theta _{5}+ \alpha _{12}^{TH}+\beta \gamma ^{TH} \vartheta ^2},\\ \Theta _{7}= \frac{\beta _{4H} \alpha _{12}^{TH}\alpha _{4H} \alpha _{22}^{TH}}{\alpha _{4H} \alpha _{21}^{TH}-\beta _{4H} \alpha _{11}^{TH}}, \\ \Theta _{8}= \frac{-\alpha _{4H}}{\alpha _{11}^{TH}\Theta _{1H}+ \alpha _{12}^{TH}},~~~~ \Theta _{9}=\frac{\alpha _{5T} \left( \alpha _{22}^{TH}-\vartheta ^2\right) -\beta _{5T}\alpha _{12}^{TH} +\beta \gamma ^{TH} \vartheta ^2}{\beta _{5T} \left( \alpha _{11}^{TH}-\gamma ^{TH} \vartheta ^2\right) -\alpha _{5T}\alpha _{21}^{TH}}, \\ \Theta _{10}=-\frac{-\alpha _{5T}}{(\alpha _{11}^{TH}-4\vartheta ^2)\Theta _{9}+ \alpha _{12}^{TH}+\beta \gamma ^{TH} \vartheta ^2},\\ \end{array} \\&\begin{array}{l} \alpha _T=2\alpha _1|q_1|^2+\alpha _2 (q_1{\overline{q}}_2+q_2{\overline{q}}_1)+2\alpha _3 |q_2|^2,~~~ \beta _T= 2\beta _1|q_1|^2+\beta _2 (q_1{\overline{q}}_2+q_2{\overline{q}}_1)+2\beta _3 |q_2|^2, \\ \alpha _H =2\alpha _1|p_1|^2+\alpha _2 (p_1{\overline{p}}_2+p_2{\overline{p}}_1)+2\alpha _3 |p_2|^2, ~~~ \beta _H= 2\beta _1|p_1|^2+\beta _2 (p_1{\overline{p}}_2p_2{\overline{p}}_1)+2\beta _3 |p_2|^2, \\ \alpha _{2T}=\alpha _1q_1 ^2 +\alpha _2 q_1 q_2 +\beta _3 q_2 ^2, \\ \beta _{2T} =\beta _1 q_1 ^2 +\beta _2 q_1 q_2 +\beta _3 q_2 ^2, ~~~ \alpha _{3T}=\alpha _T, ~~~ \beta _{3T}=\beta _T, ~~~ \alpha _{4H}=\alpha _{2T}, ~~~ \beta _{4H}=\beta _{2T}, \\ \alpha _{5T}=2\alpha _1q_1{\overline{p}}_1+\alpha _2(q_1{\overline{p}}_2+q_2{\overline{p}}_1)+2\alpha _3 q_2 {\overline{p}}_2, \\ \beta _{5T}=2\beta _1q_1{\overline{p}}_1+\beta _2(q_1{\overline{p}}_2+q_2{\overline{p}}_1)+2\beta _3 q_2 {\overline{p}}_2. \end{array} \end{aligned}$$

Now, we solve the system of \(O(\varepsilon ^3)\) term. Based on the condition of Fredholm Solubility [16]. The right-hand side of the mentioned equation has to be perpendicular to the zero eigenvectors of the adjoint operator \(M_\vartheta ^+\). At first, the zero eigenvectors of \(M_\vartheta ^+\) to the Turing bifurcation is \(q^* \exp ^{-\vartheta _j. x }\) where \(q^*= \left( q_1 ^*,q_2 ^*\right) =\left( 1,\frac{\alpha _{11} ^{TH}-\gamma \vartheta ^2}{\alpha _{21} ^{TH}}\right) ^T\). Note that \(M_\vartheta ^+\) is the conjugate of \(M_\vartheta \). Thus, the zero eigenvector of the operator \(M_\vartheta ^+\) to the Hopf bifurcation is \(p^* \exp ^{-\omega _H. t }\) where \(p^*= \left( p_1 ^*,p_2 ^*\right) ^T=\left( 1,\frac{\alpha _{11} ^{TH}}{\alpha _{21} ^{TH}}\right) ^T\). The orthogonality condition offered by \(M_\vartheta ^+\) with the third order term can e expressed as

$$\begin{aligned} \begin{array}{c} (q_1 ^*,q_2 ^* )\exp ^{-\vartheta _j r}M_\vartheta \left( {\begin{array}{c}w_3 \\ n_3\end{array}}\right) =0, ~~~ (p_1 ^*,p_2 ^* )\exp ^{-i\omega _H t}M_\vartheta \left( {\begin{array}{c}w_3 \\ n_3\end{array}}\right) =0 . \end{array} \end{aligned}$$

Then we get (by using the Fredholm solubility condition)

$$\begin{aligned} \tau _0 \frac{\partial P_1}{\partial \kappa }={\overline{\mu }} P_1 +C (\overline{n_2 P_3}+ \overline{n_3 P_2} )+ \left( g_1'|P_1|^2 +g_2 ' (|P_2|^2 +|P_3|^2)\right) P_1+g_3' |\Phi |^2 P_1, \end{aligned}$$

such that \(\tau _0,{\overline{\mu }},C,g_1 ',g_2 ',g_3 '\) are defined in “Appendix” section.

The equations of \(P_2\) and \(P_3\) could be obtained using the subscript change of P. Using the above expressions and

$$\begin{aligned} \begin{array}{l} A_j=\left( {\begin{array}{c}A^w _j\\ A^n _j\end{array}}\right) =\left( {\begin{array}{c}q_1\\ q_2\end{array}}\right) \left( \varepsilon P_j + \varepsilon ^2 P_j+ O(\varepsilon ^3) \right) , \\ B_j=\left( {\begin{array}{c}B^w _j\\ B^n _j\end{array}}\right) =\left( {\begin{array}{c}p_1\\ p_2\end{array}}\right) \varepsilon B,\\ \frac{\partial A^w _j}{\partial t }=\varepsilon ^3q_1 \frac{\partial P _j}{\partial \kappa }+ O(\varepsilon ^4), ~~~~ j=1,2,3,\\ \frac{\partial B^w _j}{\partial t }=\varepsilon ^3p_1 \frac{\partial B}{\partial \kappa }+ O(\varepsilon ^4), \end{array} \end{aligned}$$

then the amplitude equations at Turing mode (three equations) and Hopf bifurcation modes (one equation) with the orthogonal wave vector are expressed as follows (see also [15]):

$$\begin{aligned} \begin{array}{l} \frac{\partial A^w _1}{\partial t }=\mu A_1 ^w +h \overline{A_2 ^w A_3 ^w } + \left( g_1 |A_1 ^w|^2 + g_2\left( |A_2 ^w| ^2 + |A_3 ^w| ^2\right) \right) A_1 ^w +g_3|B ^w| ^2 A_1 ^w, \\ \frac{\partial A^w _2}{\partial t }=\mu A_2 ^w+h \overline{A_3 ^w A_1 ^w }+\left( g_1 |A_2 ^w|^2 + g_2\left( |A_1 ^w| ^2 + |A_3 ^w| ^2\right) \right) A_2 ^w +g_3|B ^w| ^2 A_2 ^w, \\ \frac{\partial A^w _3}{\partial t }=\mu A_3 ^w+h \overline{A_1 ^w A_2 ^w }+\left( g_1 |A_3 ^w|^2 + g_2\left( |A_1 ^w| ^2 + |A_2 ^w| ^2\right) \right) A_3 ^w +g_3|B ^w| ^2 A_3 ^w,\\ \frac{\partial B^w _1}{\partial t }=\xi B^w +g_{01}|B^w|^2H^w +g_{02}(|A_1 ^w| ^2+|A_2 ^w| ^2+|A_3 ^w| ^2)B^w,\\ \end{array}\nonumber \\ \end{aligned}$$
(3.15)

where \(h,\mu ,\xi , g_{01}, g_{02}, g_{1}, g_{2}, g_{3} \) are defined at the “Appendix” section. For the subject of determining the patterns selections, we need to analyze the existence and the stability of equilibrium points to the amplitude Eq. (3.15). In fact, each amplitude could be decomposed into its module as \(\psi _j=|A^w _j|, ~~~ \varrho = |B|\). By a straightforward calculation, we get

$$\begin{aligned} \begin{array}{l} \frac{\partial \psi _1}{\partial t }= \mu \psi _1+h \psi _2 \psi _3 \cos \nu +\left( g_1 |\psi _1 ^2|+ g_2 |\psi _2 ^2|+ g_3 |\psi _3 ^2| \right) \psi _1 +g_3 |B|^2 \psi _1,\\ \frac{\partial \psi _2}{\partial t }= \mu \psi _2+h \psi _1 \psi _3 \cos \nu +\left( g_1 |\psi _2 ^2|+ g_2 |\psi _1 ^2|+ g_3 |\psi _3 ^2| \right) \psi _2 +g_3 |B|^2 \psi _2,\\ \frac{\partial \psi _3}{\partial t }= \mu \psi _1+h \psi _1 \psi _2 \cos \nu +\left( g_1 |\psi _3 ^2|+ g_2 |\psi _2 ^2|+ g_3 |\psi _1 ^2| \right) \psi _3 +g_3 |B|^2 \psi _3,\\ \frac{\partial \varrho }{\partial t }=-h \frac{\psi _1 ^2 \psi _2 ^2+\psi _1 ^2 \psi _3 ^2+ \psi _2 ^2 \psi _3 ^2}{\psi _1 \psi _2 \psi _3 } \sin \nu , \\ \frac{\partial B}{\partial t }=\xi B + \mathfrak {R}(g_{01})|B|^2 B +\mathfrak {R}(g_{02}) (|\psi _1|^2+|\psi _2|^2+|\psi _3|^2)B,\\ \triangle \omega =\mathfrak {I}(g_{01})|B|^2+\mathfrak {I}(g_{02})(|\psi _1|^2+|\psi _2|^2+|\psi _3|^2), \end{array}\nonumber \\ \end{aligned}$$
(3.16)

where \(\nu =\sum _{j=1}^3\arg (A_j ^w)\) and \(H^w=B\exp ^{i \triangle \omega t}\). For \(h>0 \) the solution \(\varrho =0\) of the fourth equation of (3.16) is stable, and the solution \(\varrho =\pi \) is stable under the condition \(h<0\). The patterns exist only when the solution of the fourth equation of (3.16) is stable. Thus, the mode equations have the following structure:

$$\begin{aligned} \begin{array}{l} \frac{\partial \psi _1}{\partial t }= \mu \psi _1+|h| \psi _2 \psi _3 +\left( g_1 |\psi _1 ^2|+ g_2 |\psi _2 ^2|+ g_3 |\psi _3 ^2| \right) \psi _1 +g_3 |B|^2 \psi _1,\\ \frac{\partial \psi _2}{\partial t }= \mu \psi _2+|h| \psi _1 \psi _3 +\left( g_1 |\psi _2 ^2|+ g_2 |\psi _1 ^2|+ g_3 |\psi _3 ^2| \right) \psi _2 +g_3 |B|^2 \psi _2,\\ \frac{\partial \psi _3}{\partial t }= \mu \psi _1+|h| \psi _1 \psi _2 +\left( g_1 |\psi _3 ^2|+ g_2 |\psi _2 ^2|+ g_3 |\psi _1 ^2| \right) \psi _3 +g_3 |B|^2 \psi _3,\\ \frac{\partial B}{\partial t }=\xi B + \mathfrak {R}(g_{01})|B|^2 B +\mathfrak {R}(g_{02}) (|\psi _1|^2+|\psi _2|^2+|\psi _3|^2)B, \end{array}\nonumber \\ \end{aligned}$$
(3.17)
Case 1.:

The solution of spatiotemporal periodic patterns is investigated, this means that

$$\begin{aligned} \psi _1=\sqrt{\frac{\xi g_3 -\mu \mathfrak {R}(g_{01})}{g_1 \mathfrak {R}(g_{01})-g_3\mathfrak {R}(g_{02})}}, ~~~ \psi _2 =\psi _3 =0, ~~~ B= \sqrt{\frac{-\xi g_1 +\mu \mathfrak {R}(g_{02})}{g_1 \mathfrak {R}(g_{01})-g_3\mathfrak {R}(g_{02})}}. \end{aligned}$$

For having the stability with respect to the mode \(\psi _1\) and B the condition \(g_1 \mathfrak {R}(g_{01})>g_3\mathfrak {R}(g_{02})\) is required. The condition at which the solution exists reduces to \(\mu \mathfrak {R}(g_{02})\ge \xi g_1, ~~~ \xi g_3 \ge \mu \mathfrak {R}(g_{01}) .\)

Case 2.:

Considers \(\psi _1=\psi _2=\psi _3=\psi ,\) and

$$\begin{aligned} \psi =\frac{|h|+\sqrt{|h|^2-4\left( g_1+2g_2-3\frac{g_3 \mathfrak {R}(g_{02})}{\mathfrak {R}(g_{01})}\right) \left( \mu - \frac{\xi g_3}{\mathfrak {R}(g_{01})}\right) }}{2\left( g_1+2g_2-3\frac{g_3 \mathfrak {R}(g_{02})}{\mathfrak {R}(g_{01})}\right) }, ~~~ |B|^2=\frac{\xi + 3\mathfrak {R}(g_{02}) |\psi |^2}{\mathfrak {R}(g_{01})}. \end{aligned}$$

In this case the hexagonal patterns arise. For ensuring there existence and stability, the following conditions must be hold

$$\begin{aligned} |h|^2-4\left( g_1+2g_2-3\frac{g_3 \mathfrak {R}(g_{02})}{\mathfrak {R}(g_{01})}\right) \left( \mu - \frac{\xi g_3}{\mathfrak {R}(g_{01})}\right)>0, ~~~ g_1+2g_2>3\frac{g_3 \mathfrak {R}(g_{02})}{\mathfrak {R}(g_{01})}, \\ \frac{\xi + 3\mathfrak {R}(g_{02}) |\psi |^2}{\mathfrak {R}(g_{01})}>0, ~~~ 2\mathfrak {R}(g_{01})|B|^2+3|h|\psi + 6(g_1 + 2g_2) \psi ^2 <0. \end{aligned}$$

For returning back to the Turing bifurcation only it is sufficient to consider \(B=0\) where this case is investigated in many papers as it has been highlighted in the introduction section. For the Turing-Hopf bifurcation, the cases 1 and 2 show this spatiotemporal behavior (nonhomogeneous) generated by the presence of this type of co-dimensional bifurcation.

4 Numerical simulations

In this section, we will investigate the complex dynamics introduced by varying the rainfall and internal competition and the diffusion coefficient. Firstly, we discuss the method for choosing parameters based on real-world remarks. For the vegetation model (1.5), there exists a considerable interval of selecting the value of the model parameters. For instance, in the case of the rainfall constant, in our planet, there is a vast diversity in the quantity of the rainfall, which depends on the studied region, where the most significant value will be obtained in the tropic regions. In these areas, the quantity of the rainfall is very high, contrary to the desert regions, where the rain is very scarce, this gives us a very open choice of the rainfall rate for satisfying our theoretical applications. There is a quiet type of plants that can survive in a certain type of conditions of the scarcity of rain, we mention as example Opuntia ficus-indica, zebra cactus plant, golden barrel plant, and others. Besides, for the choice of the rate \(\delta \) which represents the death rate of the plant which is affected by the type of plant, as for example, the death rate for a plant is very small for the trees, and some desert plants are very small compared to the grass (see [13]). For the plant’s internal competition rate, there is also a wide interval that we can choose our parameters. In fact, this rate depends on the type of plants that can live in such as environment (where the type of environment is controlled by the rainfall constant) the reason for considering the kind of plant is due to the average of the consumed water by a plant, the length of the roots, the quantity of the consumed water by a plant (a type of the plant) and other factors which affect on determining the value of the internal competition rate. To mention that this last new parameter has been estimated in the numerical simulation. Also, the influence of this rate on the spatiotemporal dynamics is also established where it will be discussed later. In [13] it has been considered some value on the model (1.1) parameters that can be used here.

Fig. 1
figure 1

The influence of the rainfall rate R on some critical parameters of the system (1.5)

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Fig. 2
figure 2

The influence of the internal competition between plants \(\tau \) on some critical parameters of the system (1.5)

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Fig. 3
figure 3

The influence of the internal competition between plants \(\tau \) on \(D(\vartheta )\) Turing instability

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  • Fig. 1: Shows the impact of the rainfall rate on the positive homogenous steady state for the vegetation biomass \(n_+\) and on the critical value of the Turing-Hopf bifurcation defined in (3.10). The following values are considered \(\beta =0.7;~~ \delta =1.2;~~\tau =0.1;~~R_{\min }=2.5171;~~R_{0,H}=3.5272.\)

  • Fig. 2: Indicates the effect of the variable \(\tau \) on positive homogenous steady state for the vegetation biomass \(n_+\) and the impact on the critical value of the Turing-Hopf bifurcation defined in (3.10). The following values are taken: \( \beta =0.7;~~ \delta =2.2;~~R=0.1.\)

  • Fig. 3: These figure shows that \(D_\vartheta \) can be affected by the rainfall and the internal competition between plants, which means that it can affect the existence of Turing instability. The following values are chosen: \(\gamma =0.1;~~ \beta =0.7;~~ \delta =1.2.\) In fact, for the left hand multi figure values of the rainfall next to \(\tau =0.15\) are considered, and the opposite is used for the right hand figure where the rainfall rate is fixed to the value \(R=3\), and multi values of the internal competition rate are used.

  • Fig. 4: This figure represents the existence of Turing-Hopf bifurcation point which is the intersection point between the Hopf bifurcation curve defined by \(R=R_{0,H}\) (the figure in red) and Turing bifurcation curve \(\gamma =\gamma _T (1)\) defined in (3.10). The following values are chosen: \(\gamma =0.1;~~ \beta =0.7;~~ \delta =1.2; ~~ \tau =0.15.\)

  • Fig. 5, 6: These figures show the influence of \(\gamma \) on the spatiotemporal behavior of the solution of the vegetation model (1.5) where \(\beta =0.7;~~ \delta =1.2;~~\tau =0.6;~~R=4.8>R_{0,H}.\)

  • Fig. 7, 8: These figures show the influence of \(\gamma \) on the spatiotemporal behavior of the solution of the vegetation model (1.5) where \(\beta =0.7;~~ \delta =1.2;~~\tau =0.6;~~R=4.7<R_{0,H}.\)

Fig. 4
figure 4

The existence of Turing-Hopf bifurcation point which is the intersection point between the Hopf curve and Turing bifurcation curve denoted by \(T-H\)

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Fig. 5
figure 5

the effect of \(\gamma \) on the spatiotemporal behavior under the initial conditions \((w_0 (x,y),n_0 (x,y))=\left( 0.1+0.01(\cos x+\cos y),0.15+0.017(\cos x+\cos y)\right) \)

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Fig. 6
figure 6

The projection of the surfaces obtained in Fig. 5 on \(x-y\) plan

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Fig. 7
figure 7

the effect of \(\gamma \) on the spatiotemporal behavior under the initial conditions \((w_0 (x,y),n_0 (x,y))=\left( 0.1+0.01(\cos x+\cos y),0.15+0.017(\cos x+\cos y)\right) \)

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Fig. 8
figure 8

The projection of the surfaces obtained in Fig. 7 on \(x-y\) plan

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5 Discussion and remarks

In this paper, we considered a predicting vegetation model by taking into account the water restriction by the plant’s roots and the inner rivalry between the plants. Before we begin with our analysis, we recall that the inner competition between plants is just recently considered due to the paper [37], where it has been noticed that the model (1.3) cannot undergo Turing instability. Further, the main interest in the mentioned paper is to determine the patterns generated by the presence of the slope of the terrain next to the influence of internal rivalry between plants. Furthermore, there is no discussion on the types of bifurcation that the model can undergo. Based on existing works in the literature (vegetation patterns) such as the works [11, 36] we can highlight that the patterns generated by the presence of Turing bifurcation is the most common for many papers. In this context, we achieved some new results for the vegetation results for determining vegetation patterns which are elaborated by the presence of Hopf bifurcation for the non-spatial model (Theorem 3.1) wherein the paper [37] the local stability is discussed only. Further, the spatial Hopf bifurcation has never been studied for the vegetation model wherein Theorem 3.2 discuses this result in detail. To highlight that proving the existence of Hopf bifurcation for a spatial model is very important for predicting the seasonality effect on vegetation growth. Turing bifurcation is recently investigated due to the paper [11] where it has been proved that the system (1.2) (in the absence of slop of terrain) undergoes Turing bifurcation and did an excellent job in discussing the patterns in the neighborhood of the Turing bifurcation point. The presence of this result means in the actual world the possibility of existence of the vegetation cover with non equal distribution in the space,and this result is the most common in the real-world. To mention that the model studied by [11] is a particular case of our system (1.5). Turing bifurcation it self generates an important and complex patterns, which highlight the non homogeneous distribution of species, where it is applied in different disciplines, where for more information we refer to the readers to check the papers [47,48,49] The study is not confined to prove the existence of these two types of bifurcations (Hopf bifurcation and Turing bifurcation), but it is expanded to show the possibility of the intermingling of both Turing and Hopf bifurcations for different wavenumbers (0 and \(\vartheta _T\)). It is proved that Turing-Hopf bifurcation generates very complex dynamics in the case of predator-prey models [4, 9, 14, 29], mussel-algae model [30], but has never been applied to vegetation model, and no one can deny its importance. Further, the existence of Turing-Hopf bifurcation in the case of two-dimensional space is a very new step where some new arguments are provided for determining this kind of bifurcation.

Indeed, we investigated the influence of the type of climate measured by the rainfall constant R on the spatial patterns wherein Fig. 1 shows this result. More precisely, for the left-hand figure, it shows that \(\gamma _T\) defined in (3.10) is decreasing in R for different values of the wavenumber \(\vartheta \ne 1,0\) and increasing for \(\vartheta = 1\) (the figure in the middle) which shows the significant influence of these rates on the critical value of the Turing-Hopf bifurcation. Further, it has been noticed that the rainfall has a positive impact on the vegetation equilibrium state \(n_+\) as it has been shown in the left-hand of Fig. 1 which agrees with the real-life situations. In Fig. 2, the influence of the internal competition between plants on the critical values that we obtained in our analysis, as \(\gamma _T\) for different wave numbers \(\vartheta \ne 0\) in the left-hand figure where a varying impact has been remarked on the critical value of Turing instability \(\gamma _T\) and positive impact on Hopf bifurcation (the figure in the middle), and negative impact on the vegetation equilibrium state \(n_+\) which agrees with the real-world situation (by increasing the competition between the plants the vegetation density decreases). Furthermore, this influence (R and \(\tau \)) has been expanded to Turing instability in Fig.  3, which allow us to claim that these two parameters influenced Turing patterns, and this shows the importance of considering the internal competition between plants on the vegetation model. Based on the obtained figures in Figs. 5, 6, 7, and 8, we can claim the complex dynamics generated by the presence of the water redistribution by the plant’s roots next to the internal competition between plants, which is very important in predicting vegetation patterns.

As one of the results obtained in this study, we can mention that the spatial Hopf bifurcation achieved in the Theorem 3.2 is very essential for portending the vegetation patterns which can explain the seasonality of the vegetation in the case of the stable periodic solutions generated by the presence of Hopf bifurcation. The Turing bifurcation is also investigated where the influence of the water distribution in the soil by the roots next to the internal competition between plants is investigated in detail which is the main objective of our study. The presence of Hopf bifurcation can represent the impact of seasonality of the vegetation growth, wherein the actual world, we can remark that there are some seasons with a low density of the vegetation cover (for instance summer season), where the water is scarce, and there are other seasons with a high density of vegetation (for instance example the spring) where we can get a high density of vegetation cover. Furthermore, The intermingling between the Hopf and Turing bifurcation is also established for different wave numbers which are known by Turing-Hopf bifurcation, where an important agreement between the mathematical and the observation is noticed. In the actual world, it can present the possibility of having the seasonality effect generated by the presence of Hopf bifurcation and the nonhomogeneous distribution generated by the presence of Turing bifurcation. These results show the agreement between the mathematical findings and ecological observations.