A Discrete-Time Model for Consumer–Resource Interaction with Stability, Bifurcation and Chaos Control

Abstract

Keeping in mind the interactions between budmoths and the quality of larch trees located in the Swiss Alps (a mountain range in Switzerland), a discrete-time model is proposed and studied. The novel model is proposed with implementation of a nonlinear functional response that incorporates plant quality. The proposed functional response is validated with real observed data of larch budmoth interactions. Furthermore, we investigate the qualitative behavior of the proposed discrete-time system with interactions between budmoths and the quality of larch trees. Proofs of the boundedness of solutions, and the existence of fixed points and their topological classification are carried out. It is proved that the system experiences period-doubling bifurcation at its positive fixed point using the center manifold theorem and normal forms theory. Moreover, existence and direction for the torus bifurcation are also investigated for larch budmoth interactions. Bifurcating and fluctuating behaviors of the system are controlled through utilization of chaos control strategies. Numerical simulations are presented to demonstrate the theoretical findings. At the end, theoretical investigations are validated with field and experimental data.

Appendix A: Calculation Related to PDB

$$\begin{aligned} m_{13}= & {} -\frac{r_0 (2 k-r_0 x^*)}{2 k^2},\ m_{14}=\frac{a r_0 (k-r_0 x^*)}{k (a+y^*)^2},\ m_{15}=\frac{a r_0 x^* (a r_0-2 a-2 y^*)}{2 (a+y^*)^4},\\ m_{16}= & {} \frac{r_0^2 (3 k-r_0 x^*)}{6 k^3},\ m_{17}=-\frac{a r_0^2 (2 k-r_0 x^*)}{2 k^2 (a+y^*)^2},\ m_{18}=\frac{a r_0 (a r_0-2 a-2 y^*) (k-r_0 x^*)}{2 k (a+y^*)^4},\\ m_{19}= & {} \frac{1}{3} x^* \left( \frac{a^3 r_0^3}{2 (a+y^*)^6}-\frac{3 a^2 r_0^2}{(a+y^*)^5}+\frac{3 a r_0}{(a+y^*)^4}\right) ,\\ m_1= & {} \frac{a x^* (r_0 x^*-2 k)+y^* \left( k^2-k (r_0+2) x^*+r_0 x^{*2}\right) }{k^2 (a+y^*)},\\ m_2= & {} \frac{a x^* (a (k-r_0 x^*)+y^* (k r_0+k-r_0 x^*))}{k (a+y^*)^3},\\ m_3= & {} \frac{y^* \left( -2 k^2 (r_0+1)+k r_0 (r_0+4) x^*-r_0^2 x^{*2}\right) -a \left( 2 k^2-4 k r_0 x^*+r_0^2 x^{*2}\right) }{2 k^3 (a+y^*)},\\ m_5= & {} \frac{a \left( a \left( k^2-3 k r_0 x^*+r_0^2 x^{*2}\right) +y^* \left( k^2 (r_0+1)-k r_0 (r_0+3)x^*+r_0^2 x^{*2}\right) \right) }{k^2 (a+y^*)^3},\\ m_4= & {} \frac{a x^* (a (r_0-1)-y^*)}{(a+y^*)^4},\ m_{23}=\frac{c d}{(c+x^*)^3},\ m_{24}=-\frac{c d}{(c+x^*)^4}. \end{aligned}$$

$$\begin{aligned} \phi _1\left( x,y,\bar{r}\right)= & {} -\left( {\frac{m_{{13}}m_{{11}}+m_{{23}}m_{{12}}-m_{{13}}\mu _{{2}}}{m_{{12}} \left( \mu _{{2}}+1 \right) }}\right) \,{p}^{2}+\left( {\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{14}}}{m_{{12}} \left( \mu _{{2}}+1 \right) }}\right) p q\\&+ \left( {\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{15}}}{m_{{12}} \left( \mu _{{2}}+1 \right) }}\right) q^2\\&-\left( \frac{m_{{11}}m_{{16}}+m_{{12}}m_{{24}}-m_{{16}}\mu _{{2}}}{m_{{12}} \left( \mu _{{2}}+1 \right) }\right) {p}^{3}+ \left( {\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{17}}}{m_{{12}} \left( \mu _{{2}}+1 \right) }}\right) {p}^{2}q\\&+\left( {\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{18}}}{m_{{12}} \left( \mu _{{2}}+1 \right) }}\right) p q^2\\&+\left( {\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{19}}}{m_{{12}} \left( \mu _{{2}}+1 \right) }}\right) q^3+\left( {\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{1}}}{m_{{12}} \left( \mu _{{2}}+1 \right) }}\right) p\bar{r}+\left( {\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{2}}}{m_{{12}} \left( \mu _{{2}}+1 \right) }}\right) q\bar{r}\\&+\left( {\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{3}}}{m_{{12}} \left( \mu _{{2}}+1 \right) }}\right) \bar{r}p^2+\left( {\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{4}}}{m_{{12}} \left( \mu _{{2}}+1 \right) }}\right) \bar{r}q^2 +\left( {\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{5}}}{m_{{12}} \left( \mu _{{2}}+1 \right) }}\right) \bar{r}pq\\&+O\left( (|x|+|y|+|\bar{r}|)^4\right) ,\\ \psi _1\left( x,y,\bar{r}\right)= & {} \left( {\frac{ \left( 1+m_{{11}} \right) m_{{13}}}{m_{{12}} \left( \mu _{{2}} +1 \right) }}+{\frac{m_{{23}}}{\mu _{{2}}+1}}\right) \,{p}^{2}+\left( {\frac{ \left( 1+m_{{11}} \right) m_{{14}}}{m_{{12}} \left( \mu _{{2}} +1 \right) }}\right) p q+ \left( {\frac{ \left( 1+m_{{11}} \right) m_{{15}}}{m_{{12}} \left( \mu _{{2}} +1 \right) }}\right) q^2\\&+\left( {\frac{ \left( 1+m_{{11}} \right) m_{{16}}}{m_{{12}} \left( \mu _{{2}} +1 \right) }}+{\frac{m_{{24}}}{\mu _{{2}}+1}}\right) {p}^{3}+ \left( {\frac{ \left( 1+m_{{11}} \right) m_{{17}}}{m_{{12}} \left( \mu _{{2}} +1 \right) }}\right) {p}^{2}q+\left( {\frac{ \left( 1+m_{{11}} \right) m_{{18}}}{m_{{12}} \left( \mu _{{2}} +1 \right) }}\right) p q^2\\&+\left( {\frac{ \left( 1+m_{{11}} \right) m_{{19}}}{m_{{12}} \left( \mu _{{2}} +1 \right) }}\right) q^3+\left( {\frac{ \left( 1+m_{{11}} \right) m_{{1}}}{m_{{12}} \left( \mu _{{2}}+ 1 \right) }}\right) p\bar{r}+\left( {\frac{ \left( 1+m_{{11}} \right) m_{{2}}}{m_{{12}} \left( \mu _{{2}}+ 1 \right) }}\right) q\bar{r}\\&+\left( {\frac{ \left( 1+m_{{11}} \right) m_{{3}}}{m_{{12}} \left( \mu _{{2}}+ 1 \right) }}\right) \bar{r}p^2+\left( {\frac{ \left( 1+m_{{11}} \right) m_{{4}}}{m_{{12}} \left( \mu _{{2}}+ 1 \right) }}\right) \bar{r}q^2 +\left( {\frac{ \left( 1+m_{{11}} \right) m_{{5}}}{m_{{12}} \left( \mu _{{2}}+ 1 \right) }}\right) \bar{r}pq\\&+O\left( (|x|+|y|+|\bar{r}|)^4\right) .\\ d_1= & {} \left( {\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{13}}}{m_{{12}} \left( \mu _{{2}}+1 \right) }}-{\frac{m_{{23}}}{\mu _{{2}}+1}} \right) m_{12}^{2}-{\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{ {14}} \left( 1+m_{{11}} \right) }{\mu _{{2}}+1}}\\&+{\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{15}} \left( 1+m_{{11}} \right) ^{2}}{m_{{12}} \left( \mu _{{2}}+1 \right) }},\\ d_2= & {} {\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{1}}}{\mu _{{2}}+1}}-{ \frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{2}} \left( 1+m_{{11}} \right) }{m_{{12}} \left( \mu _{{2}}+1 \right) }},\\ d_3= & {} {\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{12}}m_{{3}}}{\mu _{{2}}+ 1}}-{\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{5}} \left( 1+m_{{11 }} \right) }{\mu _{{2}}+1}}+{\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{1}}l_{{1}}}{\mu _{{2}}+1}}\\&+{\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{4}} \left( 1+m_{{11}} \right) ^{2}}{m_{{12}} \left( \mu _{{2}}+1 \right) }}+{\frac{ \left( \mu _{{2}}-m_{{11}} \right) ^{2}m_{{2}}l_{{1}}}{m_{{12}} \left( \mu _{{2 }}+1 \right) }}\\&+2\, \left( {\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{13}}}{m_{{12}} \left( \mu _{{2}}+1 \right) }}-{\frac{m_{{23}}}{\mu _{{2}}+1}} \right) m_{12}^{2}l_{{2}}\\&+{\frac{ \left( \mu _{{2}}-m_{{11}} \right) ^{2}m_{{14}}l_{{2}}}{\mu _{{ 2}}+1}}-{\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{14}}l_{{2}} \left( 1+m_{{11}} \right) }{\mu _{{2}}+1}}\\&-2\,{\frac{ \left( \mu _{{2} }-m_{{11}} \right) ^{2}m_{{15}} \left( 1+m_{{11}} \right) l_{{2}}}{m_{ {12}} \left( \mu _{{2}}+1 \right) }},\\ d_4= & {} {\frac{ \left( \mu _{{2}}-m_{{11}} \right) l_{{2}} \left( m_{{1}}m_{{ 12}}-m_{{2}}m_{{11}}+m_{{2}}\mu _{{2}} \right) }{m_{{12}} \left( \mu _{{ 2}}+1 \right) }}, \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} d_5&=\left( {\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{16}}}{m_{{12}} \left( \mu _{{2}}+1 \right) }}-{\frac{m_{{24}}}{\mu _{{2}}+1}} \right) m_{12}^{3}-{\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{ {12}}m_{{17}} \left( 1+m_{{11}} \right) }{\mu _{{2}}+1}}\\&\quad +2\left( {\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{13}}}{m_{{12 }} \left( \mu _{{2}}+1 \right) }}-{\frac{m_{{23}}}{\mu _{{2}}+1}} \right) m_{12}^{2}l_{{1}}+{\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{18}} \left( 1+m_{{11}} \right) ^{2}}{\mu _{{2}}+1}}\\&\quad -{\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{14}}l_{{1}} \left( 1+m_ {{11}} \right) }{\mu _{{2}}+1}}-{\frac{ \left( \mu _{{2}}-m_{{11}} \right) m_{{19}} \left( 1+m_{{11}} \right) ^{3}}{m_{{12}} \left( \mu _ {{2}}+1 \right) }}\\&\quad +{\frac{ \left( \mu _{{2}}-m_{{11}} \right) ^{2}m_{{14}}l_{{1}}}{\mu _{{2}}+1}}-{\frac{ 2\left( \mu _{{2}}-m_{{11}} \right) ^{2}m_{{15}} \left( 1+m_ {{11}} \right) l_{{1}}}{m_{{12}} \left( \mu _{{2}}+1 \right) }}. \end{aligned} \end{aligned}$$