Spatial patterns in a vegetation model with internal competition and feedback regulation

Abstract

The vegetation patterns are a characteristic particularity of semiarid zones, which can be the future of modern ecology for its importance. This paper aims to study a diffusive vegetation model for the subject of studying the complex patterns generated by the presence of Turing-Hopf bifurcation. The main focus is on analyzing the effect of the locative internal rivalry between plants and feedback regulation on the pattern formations. The cross-diffusion produced by the positive feedback regulation generates surprising dynamics such as Hopf bifurcation, Turing bifurcation, Turing-Hopf bifurcation, which confirms the imbalance of the distribution of the vegetation in the semi-desert regions. For analyzing the spatiotemporal behavior near the Turing-Hopf bifurcation point, the Amplitude equation restricted at this point is used. Further, by using numerical simulations, the complex dynamics induced by the positive feedback redistribution and inner competition are explored.

Appendix

$$\begin{aligned}&\begin{array}{l} \tau _0=q_1 q_1 ^* + q_2 q_2 ^*, ~~~~ {\overline{\mu }}= q_1 q_1 ^* \left( \alpha ^{(2)}_{11}+\gamma ^{(2)}\vartheta ^2\right) +q_1 q_2 ^* \alpha ^{(2)}_{21}\\ +q_2 q_1 ^* \left( \alpha ^{(2)}_{12}-\gamma ^{(2)}\beta \vartheta ^2\right) +q_2 q_2 ^* \alpha ^{(2)}_{22},\\ C= q_1 ^* (2\alpha _1 q_1 ^2 +\alpha _2 q_1 q_2 +2\alpha _3 q_2 ^2)+q_2 ^* (2\beta _1 q_1 ^2 +\beta _2 q_1 q_2 + 2 \beta _3 q_2 ^2 ),\\ g_1 '= \left( \Theta _{1T}\Theta _{2T}+\Theta _{3}\Theta _{4}\right) \left( q_1 ^* (2\alpha _1 q_1 + \alpha _2 q_2)+q_2 ^* (2\beta _1 q_1 +\beta _2 q_2 ) \right) \\ \quad + \left( \Theta _{2T}+\Theta _{4}\right) \left( q_1 ^* (\alpha _2 q_1 + 2\alpha _3 q_2)+q_2 ^* (\beta _2 q_1 +2\beta _3 q_2 ) \right) +3\alpha _4 q_1 ^3,\\ g_2 '= \left( \Theta _{1T}\Theta _{2T}+\Theta _{3}\Theta _{4}\right) \left( q_1 ^* (2\alpha _1 q_1 + \alpha _2 q_2)+q_2 ^* (2\beta _1 q_1 +\beta _2 q_2 ) \right) \\ \quad + \left( \Theta _{2T}+\Theta _{6}\right) \left( q_1 ^* (\alpha _2 q_1 + 2\alpha _3 q_2)+q_2 ^* (\beta _2 q_1 +2\beta _3 q_2 ) \right) +6\alpha _4 q_1 ^3,\\ g_3 '= \Theta _{1H} \Theta _{2H}\left( 2q_1 ^* (\alpha _1 q_1 + \alpha _3 q_2)+2q_2 ^* (\beta _1 q_1 +\beta _3 q_2 ) \right) \\ \quad + \Theta _{2H}( q_1 + q_2)\left( q_1 ^* \alpha _2+q_2 ^* \beta _2 \right) +6(q_1 ^* \alpha _4 +q_2 ^* \beta _4)q_1 |p_1 |^2. \end{array} \\&\begin{array}{l} h=\frac{C}{\tau _0 q_1}, ~~~ g_1= \frac{g_1 '}{\tau _0 q_1 ^2}, ~~~ g_2= \frac{g_2 '}{\tau _0 q_1 ^2} , ~~~ g_3= \frac{g_3 '}{\tau _0 p_1 ^2}, \\ \xi =\frac{q_1p_2 ^2(R-R_{H,0})\alpha ^{(1)} _{21}}{q_1 p_1 ^* +q_2 p_2 ^*},\\ \mu = q_1 q_1 ^* \left( \alpha ^{(1)}_{11} (R-R_{H,0})+(\gamma -\gamma _T ^*) \vartheta ^2\right) +q_1 q_2 ^* \alpha ^{(1)}_{21}(R-R_{H,0})\\ \quad +q_2 q_1 ^* \left( \alpha ^{(1)}_{12} (R-R_{H,0})-(\gamma -\gamma _T ^*)\beta \vartheta ^2\right) +q_2 q_2 ^* \alpha ^{(1)}_{22},\\ g_{01}=\frac{1}{(q_1 p_1 ^* +q_2 p_2 ^*)q_1 ^2}\left( p_1 ^* \left( (2 \alpha _1 {\overline{p}}_1 +\alpha _2 {\overline{p}}_2)\Theta _7 \Theta _8 \right. \right. \\ \quad \left. +( \alpha _2 {\overline{p}}_1 +2\alpha _3 {\overline{p}}_2)\Theta _8 +3 \alpha _4 |p_1|^2 p_1 \right) \\ \quad \left. +p_2 ^2 \left( (2 \beta _1 {\overline{p}}_1 +\beta _2 {\overline{p}}_2)\Theta _7 \Theta _8 \right. \right. \\ \quad \left. \left. +( \beta _2 {\overline{p}}_1 +2\beta _3 {\overline{p}}_2)\Theta _8 +3 \beta _4 |p_1|^2 p_1\right) \right) ,\\ g_{02}=\frac{1}{(q_1 p_1 ^* +q_2 p_2 ^*)q_1 ^2}\left( p_1 ^* \left( (2 \alpha _1 {\overline{q}}_1 +\alpha _2 {\overline{q}}_2)\Theta _9 \Theta _{10} +( \alpha _2 {\overline{q}}_1 +2\alpha _3 {\overline{q}}_2)\Theta _{10} +3 \alpha _4 |q_1|^2 p_1 \right) \right. \\ \quad +p_2 ^2 \left( (2 \beta _1 {\overline{q}}_1 \right. \\ \quad \left. \left. +\beta _2 {\overline{q}}_2)\Theta _9 \Theta _{10} +( \beta _2 {\overline{q}}_1 +2\beta _3 {\overline{q}}_2)\Theta _{10}+3 \beta _4 |q_1|^2 p_1\right) \right) , \end{array} \end{aligned}$$