Pest Control of Jatropha curcas Plant for Different Response Functions

Abstract

Nowadays Jatropha curcas plants are being considered as a renewable energy feedstock for the production of biodiesel to overcome the crisis of natural fuel. The seeds of this plant contain oil which is one of the significant resources for alternative fuel production i.e. biodiesel. Though Jatropha curcas plant is proven to be toxic to many insects and animals, it is not pest and disease resistant. On this regard our research article presents formulation and analysis of a mathematical model for Jatropha plantation with a view to control its natural pests through application of biological pesticide. We assume linear and hyperbolic functional responses of predator for susceptible pest, where for infected pest the functional response is linear, as infected pests are weaker than susceptible pest and easy to catch. We study the dynamics of the system around each of the ecological feasible equilibrium. The reduction of disease eradication and predator-pest coexistence are observed around the predator free and disease free equilibrium respectively.

Appendices

Appendix A

Coefficients of $A\bar{s}^3+B\bar{s}^2+C\bar{s}+D=0$ are as follows,

$A= \frac{A'}{r_j k_s \xi [\mu _v -(\kappa \lambda - \gamma )\bar{s}]}$, $B= \frac{B'}{r_j k_s \xi [\mu _v -(\kappa \lambda - \gamma )\bar{s}]}$,

$C= \frac{C'}{r_j k_s \xi [\mu _v -(\kappa \lambda - \gamma )\bar{s}]}$, $D=\frac{D'-\lambda \pi _v r_j k_s \xi }{r_j k_s \xi [\mu _v -(\kappa \lambda - \gamma )\bar{s}]}$;

where

$A'= r_s k_j \alpha \xi (\gamma - \kappa \lambda )$, $B'= r_s k_j (\alpha \xi k_s \kappa \lambda - \alpha \xi \lambda k_s + \alpha \xi \mu _v + \xi r_j \kappa \lambda - r_j \xi \gamma + \alpha \mu \lambda )$,

$C'= r_s k_j (\xi \lambda r_j k_s - \alpha \xi \mu _v k_s - \xi k_s \kappa \lambda r_j - \xi \mu _v r_j - \lambda \mu _v r_j)$, $D'= \xi \mu _v r_s r_j k_j k_s$,

The Jacobian matrix for predator free equilibrium point for type I functional response is given by,

$$J = \left( \begin{array}{ccccc} a^{11} &{} a^{12} &{} 0 &{} 0 &{} 0 \\ a^{21} &{} a^{22} &{} a^{23} &{} a^{24} &{} a^{25}\\ 0 &{} a^{32} &{} a^{33} &{} a^{34} &{} a^{35}\\ 0 &{} 0 &{} 0 &{} a^{44} &{} 0\\ 0 &{} a^{52} &{} a^{53} &{} 0 &{} a^{55} \end{array} \right) $$

where

$a^{11}=r_j-\alpha \bar{s}-2\bar{j}\frac{r_j}{k_j}$, $a^{12}=-\alpha \bar{j}$, $a^{21}=r_s\bar{s}(1-\frac{\bar{s}+\bar{i}}{k_s})$,

$a^{22}=r_s\bar{j}(1-\frac{\bar{s}+\bar{i}}{k_s})-\frac{r_s}{k_s}\bar{j}\bar{s}-\lambda \bar{v}$, $a^{23}=-\frac{r_s}{k_s}\bar{j}\bar{s}$,

$a^{24}=-\beta \bar{s}$, $a^{25}=-\lambda \bar{s}$, $a^{32}=\lambda \bar{v}$, $a^{33}=-\xi$,

$a^{34}=-l\bar{i}$, $a^{35}=\lambda \bar{s}$, $a^{44}=-d_p+\theta _1 \beta \bar{s}+\theta _2 \bar{i}$, $a^{52}=-\gamma \bar{v}$, $a^{53}=k\xi$, $a^{55}=-(\mu _v+\gamma \bar{s}).$

The coefficients of Eq. 1 are as follows,

$B_1=-\sum {a^{ii}}$,

$B_2=\sum {a^{ii}a^{jj}}-\sum {a^{ij}a^{ji}}$,

$B_3=-\sum {a^{ii}a^{jj}a^{kk}}+\sum {a^{ij}a^{ji}a^{kk}}-\sum {a^{ij}a^{jk}a^{ki}}$,

$B_4=\sum {a^{ii}a^{jj}a^{kk}a^{ll}}-\sum {a^{ij}a^{ji}a^{kk}a^{ll}}+\sum {a^{ij}a^{jk}a^{ki}a^{ll}}-\sum {a^{ij}a^{ji}a^{kl}a^{lk}}$,

$B_5=-a^{ii}a^{jj}a^{kk}a^{ll}a^{mm}+\sum {a^{ij}a^{ji}a^{kk}a^{ll}a^{mm}}-\sum {a^{ij}a^{jk}a^{ki}a^{ll}a^{nn}}+\sum {a^{ij}a^{ji}a^{kl}a^{lk}a^{mm}}$.

($i,j,k,l,m=\{1,2,3,4,5\}$ and $i\ne {j}\ne {k}\ne {l}\ne {m}$)

Appendix B

The coefficients of Eq. 5 are given as follows,

$C_1$=$\frac{l\mu _v \theta _2}{\varepsilon _p s^*}+\gamma l \theta _2$,

$C_2$=$k\lambda \xi -\mu _v\xi -\frac{l\mu _v\theta _1\beta }{\varepsilon _p}-\frac{l\mu _v d_p}{\varepsilon _p s^*}-\gamma \xi -\frac{\gamma l}{\varepsilon _p}(\theta _1\beta s^*-d_p)$,

$C_3$=$\lambda \pi _v$.

The constants in $(s-a)(s-b)>0 (a<b)$ are

$a=\frac{-(\varepsilon _p\mu _v\xi +l\mu _v\theta _1\beta +\varepsilon _p\gamma \xi -\kappa \varepsilon _p\xi -l\gamma d_p)-\surd ((\varepsilon _p\mu _v\xi +l\mu _v\theta _1\beta +\varepsilon _p\gamma \xi -\kappa \varepsilon _p\xi -l\gamma d_p)^2-4l^2\gamma \theta _1\beta \mu _v d_p)}{2l\gamma \theta _1\beta }$ and

$b=\frac{-(\varepsilon _p\mu _v\xi +l\mu _v\theta _1\beta +\varepsilon _p\gamma \xi -\kappa \varepsilon _p\xi -l\gamma d_p)+\surd ((\varepsilon _p\mu _v\xi +l\mu _v\theta _1\beta +\varepsilon _p\gamma \xi -\kappa \varepsilon _p\xi -l\gamma d_p)^2-4l^2\gamma \theta _1\beta \mu _v d_p)}{2l\gamma \theta _1\beta }$.

The Jacobian matrix for the interior equilibrium point for type I functional response is given by,

$$J = \left( \begin{array}{ccccc} a_{11} &{} a_{12} &{} 0 &{} 0 &{} 0 \\ a_{21} &{} a_{22} &{} a_{23} &{} a_{24} &{} a_{25}\\ 0 &{} a_{32} &{} a_{33} &{} a_{34} &{} a_{35}\\ 0 &{} a_{42} &{} a_{43} &{} a_{44} &{} 0\\ 0 &{} a_{52} &{} a_{53} &{} 0 &{} a_{55} \end{array} \right) $$

where,

where, $a_{11}=r_j-\alpha s^*-2j^*\frac{r_j}{k_j}$, $a_{12}=-\alpha j^*$, $a_{21}=r_ss^*(1-\frac{s^*+i^*}{k_s})$,

$a_{22}=r_sj^*(1-\frac{s^*+i^*}{k_s})-\frac{r_s}{k_s}j^*s^*-\lambda v^*-\beta p^*$, $a_{23}=-\frac{r_s}{k_s}j^*s^*$,

$a_{24}=-\beta s^*$, $a_{25}=-\lambda s^*$, $a_{32}=\lambda v^*$, $a_{33}=-(\xi +lp^*)$,

$a_{34}=-li^*$, $a_{35}=\lambda s^*$, $a_{42}=\theta _1 \beta p^*$, $a_{43}=\theta _2 p^*$,

$a_{44}=-d_p-2\varepsilon _p p^*+\theta _1 \beta s^*+\theta _2 i^*$, $a_{52}=-\gamma v^*$ $a_{53}=\kappa \xi$ $a_{55}=-(\mu _v+\gamma s^*).$

The coefficients of Eq. 6 are as follows,

$D_1=-\sum {a_{ii}}$,

$D_2=\sum {a_{ii}a_{jj}}-\sum {a_{ij}a_{ji}}$,

$D_3=-\sum {a_{ii}a_{jj}a_{kk}}+\sum {a_{ij}a_{ji}a_{kk}}-\sum {a_{ij}a_{jk}a_{ki}}$,

$D_4=\sum {a_{ii}a_{jj}a_{kk}a_{ll}}-\sum {a_{ij}a_{ji}a_{kk}a_{ll}}+\sum {a_{ij}a_{jk}a_{ki}a_{ll}}-\sum {a_{ij}a_{ji}a_{kl}a_{lk}}$,

$D_5=-a_{ii}a_{jj}a_{kk}a_{ll}a_{mm}+\sum {a_{ij}a_{ji}a_{kk}a_{ll}a_{mm}}-\sum {a_{ij}a_{jk}a_{ki}a_{ll}a_{nn}}+\sum {a_{ij}a_{ji}a_{kl}a_{lk}a_{mm}}-\sum {a_{ii}a_{jk}a_{kl}a_{lm}a_{mj}}$.

($i,j,k,l,m=\{1,2,3,4,5\}$ and $i\ne {j}\ne {k}\ne {l}\ne {m}$)

Appendix C

Coefficients of $\grave{A}\hat{s}^3+\grave{B}\hat{s}^2+\grave{C}\hat{s}+\grave{D}=0$ are as follows,

$\grave{A}= \frac{\tilde{A}}{r_j k_s \xi [\mu _v -(\kappa \lambda - \gamma )\hat{s}]}$, $\grave{B}= \frac{\tilde{B}}{r_j k_s \xi [\mu _v -(\kappa \lambda - \gamma )\hat{s}]}$,

$\grave{C}= \frac{\tilde{C}}{r_j k_s \xi [\mu _v -(\kappa \lambda - \gamma )\hat{s}]}$, $\grave{D}=\frac{\tilde{D}-\lambda \pi _v r_j k_s \xi }{r_j k_s \xi [\mu _v -(\kappa \lambda - \gamma )\hat{s}]}$;

where

$\tilde{A}= r_s k_j \alpha \xi (\gamma - \kappa \lambda )$, $\tilde{B}= r_s k_j (\alpha \xi k_s \kappa \lambda - \alpha \xi \lambda k_s + \alpha \xi \mu _v + \xi r_j \kappa \lambda - r_j \xi \gamma + \alpha \mu \lambda )$,

$\tilde{C}= r_s k_j (\xi \lambda r_j k_s - \alpha \xi \mu _v k_s - \xi k_s \kappa \lambda r_j - \xi \mu _v r_j - \lambda \mu _v r_j)$, $\tilde{D}= \xi \mu _v r_s r_j k_j k_s$,

The Jacobian matrix for the predator-free equilibrium point for type II functional response is given by,

$$ J = \left( \begin{array}{ccccc} b^{11} &{} b^{12} &{} 0 &{} 0 &{} 0 \\ b^{21} &{} b^{22} &{} b^{23} &{} b^{24} &{} b^{25}\\ 0 &{} b^{32} &{} b^{33} &{} b^{34} &{} b^{35}\\ 0 &{} 0 &{} 0 &{} b^{44} &{} 0\\ 0 &{} b^{52} &{} b^{53} &{} 0 &{} b^{55} \end{array} \right) $$

where

$b^{11}=r_j-\alpha \hat{s}-2\hat{j}\frac{r_j}{k_j}$, $b^{12}=-\alpha \hat{j}$, $b^{21}=r_s\hat{s}(1-\frac{\hat{s}+\hat{i}}{k_s})$,

$b^{22}=r_s\hat{j}(1-\frac{\hat{s}+\hat{i}}{k_s})-\frac{r_s}{k_s}\hat{j}\hat{s}-\lambda \hat{v}$, $b^{23}=-\frac{r_s}{k_s}\hat{j}\hat{s}$,

$b^{24}=-\beta \frac{\hat{s}}{\alpha +\hat{s}}$, $b^{25}=-\lambda \hat{s}$, $b^{32}=\lambda \hat{v}$, $b^{33}=-\xi$,

$b^{34}=-l\hat{i}$, $b^{35}=\lambda \hat{s}$, $b^{44}=-d_p+\theta _1 \beta \frac{\hat{s}}{\alpha +\hat{s}}+\theta _2 \hat{i}$, $b^{52}=-\gamma \hat{v}$, $b^{53}=\kappa \xi$, $b^{55}=-(\mu _v+\gamma \hat{s}).$

The coefficients of Eq.  9 are as follows,

$b_1=-\sum {b^{ii}}$,

$b_2=\sum {b^{ii}b^{jj}}-\sum {b^{ij}b^{ji}}$,

$b_3=-\sum {b^{ii}b^{jj}b^{kk}}+\sum {b^{ij}b^{ji}b^{kk}}-\sum {b^{ij}b^{jk}b^{ki}}$,

$b_4=\sum {b^{ii}b^{jj}b^{kk}b^{ll}}-\sum {b^{ij}b^{ji}b^{kk}b^{ll}}+\sum {b^{ij}b^{jk}b^{ki}b^{ll}}-\sum {b^{ij}b^{ji}b^{kl}b^{lk}}$,

$b_5=-b^{ii}b^{jj}b^{kk}b^{ll}b^{mm}+\sum {b^{ij}b^{ji}b^{kk}b^{ll}b^{mm}}-\sum {b^{ij}b^{jk}b^{ki}b^{ll}b^{nn}}+\sum {b^{ij}b^{ji}b^{kl}b^{lk}b^{mm}}$.

($i,j,k,l,m=\{1,2,3,4,5\}$ and $i\ne {j}\ne {k}\ne {l}\ne {m}$)

Appendix D

The coefficients of Eq. 11 are as follows,

$c_1=\pi _v\lambda \varepsilon _p r_j^2+\kappa \xi \lambda \varepsilon _p r_j^2-\gamma \xi \varepsilon _pr_j^2i_*-\gamma \theta _1\beta r_j^2+\gamma \theta _2\alpha k_j a r_j-\gamma \theta _2 r_ji_*$,

$c_2=-\pi _v\lambda r_j\varepsilon _pa\alpha k_j-\kappa \xi \lambda r_j \varepsilon _pa\alpha k_ji_*-2\pi _v\lambda r_j^2\varepsilon _p k_j-2\kappa k_j\xi \lambda r_j^2\varepsilon _p+\\ \xi \mu _v\alpha k_jr_j\varepsilon _pi-\mu _v\alpha k_jld_pr_ji_*+\theta _1\beta r_j+\theta _2i_*+\gamma \xi \varepsilon _p a\alpha r_jk_ji_*+2\gamma \xi \varepsilon _pr_j^2k_ji_*-\gamma r_jd_pal\alpha r_ji_*-\\ \gamma r_j^2d_pk_jli_*+\gamma r_j^2\theta _1\beta k_j+\gamma r_jk_j\theta _1\beta +\gamma r_jk_j^2\theta _2a\alpha +\gamma r_j^2\theta _2k_ji_*+\\ \gamma r_jk_j\theta _2i_*,$

$c_3=\pi _v\lambda r_j\varepsilon _pa\alpha k_j^2+\kappa k_j^2\xi \lambda r_j^2\varepsilon _p+\kappa \xi \lambda r_j\varepsilon _pa\alpha k_j^2i_*-\xi \mu _v\alpha ^2k_j^2\varepsilon _p-\xi \mu _v\alpha k_j^2r_j\varepsilon _pi_*\\ +\mu _v\alpha ^2k_j^2ld_pai_*+\mu _v\alpha k_j^2ld_pr_ji_*-\theta _1\beta r_jk_j-\theta _2a\alpha k_ji_*-\theta _2k_jr_ji_*-\\ \lambda \xi \varepsilon _pa\alpha r_jk_j^2i_*-\gamma \xi \varepsilon _p r_j^2k_j^2i_*+\gamma r_jk_j^2d_pal\alpha i_*+\gamma r_j^2k_j^2d_pli_*-\gamma r_j^2k_j^2\theta _1\beta - \\ \gamma r_j^2k_j^2\theta _2i_*.$

The Jacobian matrix for the interior equilibrium point for type II functional response is given by,

$$ J = \left( \begin{array}{ccccc} b_{11} &{} b_{12} &{} 0 &{} 0 &{} 0 \\ b_{21} &{} b_{22} &{} b_{23} &{} b_{24} &{} b_{25}\\ 0 &{} b_{32} &{} b_{33} &{} b_{34} &{} b_{35}\\ 0 &{} b_{42} &{} b_{43} &{} b_{44} &{} 0\\ 0 &{} b_{52} &{} b_{53} &{} 0 &{} b_{55} \end{array} \right) $$

where,

$b_{11}=r_j-\alpha s^*-2j^*\frac{r_j}{k_j}$, $b_{12}=-\alpha j^*$, $b_{21}=r_ss^*(1-\frac{s^*+i^*}{k_s})$,

$b_{22}=r_sj^*(1-\frac{s^*+i^*}{k_s})-\frac{r_s}{k_s}j^*s^*-\lambda v^*-\beta p^*$, $b_{23}=-\frac{r_s}{k_s}j^*s^*$,

$b_{24}=-\beta s^*$, $b_{25}=-\lambda s^*$, $b_{32}=\lambda v^*$, $b_{33}=-(\xi +lp^*)$,

$b_{34}=-li^*$, $b_{35}=\lambda s^*$, $b_{42}=\theta _1 \beta p^*$, $b_{43}=\theta _2 p^*$,

$b_{44}=-d_p-2\varepsilon _p p^*+\theta _1 \beta s^*+\theta _2 i^*$, $b_{52}=-\gamma v^*$, $b_{53}=\kappa \xi$, $b_{55}=-(\mu _v+\gamma s^*)$.

The coefficients of Eq. 12 are as follows,

$\grave{B_1}=-\sum {b_{ii}}$,

$\grave{B_2}=\sum {b_{ii}b_{jj}}-\sum {b_{ij}b_{ji}}$,

$\grave{B_3}=-\sum {b_{ii}b_{jj}b_{kk}}+\sum {b_{ij}b_{ji}b_{kk}}-\sum {b_{ij}b_{jk}b_{ki}}$,

$\grave{B_4}=\sum {b_{ii}b_{jj}b_{kk}b_{ll}}-\sum {b_{ij}b_{ji}b_{kk}b_{ll}}+\sum {b_{ij}b_{jk}b_{ki}b_{ll}}-\sum {b_{ij}b_{ji}b_{kl}b_{lk}}$,

$\grave{B_5}=-b_{ii}b_{jj}b_{kk}b_{ll}b_{mm}+\sum {b_{ij}b_{ji}b_{kk}b_{ll}b_{mm}}-\sum {b_{ij}b_{jk}b_{ki}b_{ll}b_{nn}}+\sum {b_{ij}b_{ji}b_{kl}b_{lk}b_{mm}}-\sum {b_{ii}b_{jk}b_{kl}b_{lm}b_{mj}}$.

($i,j,k,l,m=\{1,2,3,4,5\}$ and $i\ne {j}\ne {k}\ne {l}\ne {m}$)