Theoretical Assessment of the Impact of Water Stress on Plants Production: Case of Banana-Plantain

Abstract

The aim of this paper is to investigate the role of water stress on plants production. We propose a mathematical model for the dynamics growth of plants that takes into account the concentration of available water in the soil, water stress, plant production and plants compensation. Sensitivity analysis of the model has been performed in order to determine the impact of related parameters on the dynamics growth of plants. We present the theoretical analysis of the model with and without water stress. More precisely, we show that the full model is well-posedness. For each model, we compute the trivial equilibria and derive two thresholds parameters that determine the outcome of water stress within a plantation. Further, we perform numerical simulation on the case of banana-plantain simulations to support the theory. We found that the Hopf bifurcation occurs for a specific value of the water absorption rate of unstressed plants. The impact of the water stress on the banana-plantain production is also numerically investigated. After, the role of the water stress on the plant production is numerically investigated. We found that the water stress can cause about 68.16% of loss of banana-plantain production within a plantation with 1600 rejets initially planted. This suggests that climate change plays a detrimental role on banana-plantains production.

Appendices

Appendix A: Proof of Lemma 3.

Here, we prove Lemma 3.

Using the first equation of system (9), one has that

$$\begin{aligned} {\dot{E}}_s(t)&= r_1E_s(t)(1-\dfrac{E_s(t)}{K_1})-a_1(P_v(t)+P_M(t)+P_s(t))E_s(t){ V(t)}- \mu _1E_s(t),\\&\le r_1E_s(t)(1-\dfrac{E_s(t)}{K_1})-\mu _1E_s(t),\\&\le (r_1-\mu _1)E_s(t)-\dfrac{r_1}{K_1}(E_{s}(t))^{2}. \end{aligned}$$

Let

$$\begin{aligned} \dot{M(t)}=(r_1-\mu _1)M(t)-\dfrac{r_1}{K_1}M^{2}(t). \end{aligned}$$

Solving the above equation gives

$$\begin{aligned} M(t)=\dfrac{K_1(r_1-\mu _1)}{r_1+K_1(r_1-\mu _1)C_1\exp (-(r_1-\mu _1))}, \end{aligned}$$

where $C_1$ is gives in Eq. (17). This implies that $\forall t\ge 0$

$$\begin{aligned} M(t)\le \dfrac{K_1(r_1-\mu _1)}{r_1} = K_1\left( 1-\dfrac{\mu _1}{r_1}\right) =E_{s}^{0} \end{aligned}$$

Using the comparison theorem (SMITH et al. 1995), one has $E_{s}(t)\le K_1\left( 1-\dfrac{\mu _1}{r_1}\right) =E_{s}^{0}$ for all $t\ge 0$ since $E_{s}(0)=M(0)$, $\forall t\ge 0$.

Then, for system (9), one can conclude that $E_s\le E_{s}^{0}$ for all $t\ge 0$. This concludes the proof. $\square$

Appendix B: Comparison Theorem

In this Appendix, we recall the comparaison theorem that has been used to prove the global asymptotic stability of the trivial equilibrium point $Q_{0}^{(2)}$ of system (9).

Theorem 6

Marek (1970); Burlando (1991) Let E be an ordered Banach space with normal and generating positive cone $C_p$ and let $A_1$ and $A_2$ be positive operators in L(E, E) ( the space of all linear and continuous operators on E) such that $A_1 \le A_2$. Then $\rho (A_1) \le \rho (A_2)$.

Appendix C: Proof of Theorem 3

In this appendix, we give the proof of Theorem 3 on the local asymptotic stability of the non trivial equilibrium equilibrium point $Q^*$ of system (9). In order to apply the theorem of Castillo-Chavez and Song (2004), the following simplification and change of variables are first of all made. Let $(z_1, z_2, z_3) = (E_s, P_v, P_M)$. Then, system (9) can be written in the following compact form:

$$\begin{aligned} {\dot{z}}=f(z), \end{aligned}$$

with $f=(f_1, f_2, f_3)$ as follows:

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{z_1} ~=~ r_1\left( z_1+E_{s}^{0}\right) \left( 1-\dfrac{z_1}{K_1}\right) -a_1(z_2+z_3)\left( z_1+E_{s}^{0}\right) - \mu _1\left( z_1+E_{s}^{0}\right) =f_1,\\ \dot{z_{2}} ~=~r_2ea_{1}\left( z_1+E_{s}^{0}\right) z_3\left( 1-\dfrac{z_2}{K_2}\right) -(\alpha _1+\mu _2)z_2=f_2,\\ \dot{z_{3}} ~=~\alpha _1z_2-\mu _3z_3=f_3. \end{array} \right. \end{aligned}$$

(39)

The Jacobian of system (39) at $Q_{0}^{(2)}$ is

$J(Q_{0}^{(2)})= \left( \begin{array}{cccccc} \mu _1-r_1&{} -a_1E_{s}^{0}&{} -a_1E_{s}^{0}\\ 0&{} -(\alpha _1+\mu _2)&{} r_2ea_{1}E_{s}^{0}\\ 0&{} \alpha _1&{} -\mu _3 \\ \end{array}\right) ,$

It follows that the Jacobian $J(Q_{0}^{(2)})$ of system (39) at the trivial equilibrium point $Q_{0}^{(2)}$, with $\alpha _1 = \alpha _1^*$, denoted by $J_{\alpha _1^*}$, has a simple zero eigenvalue (with all other eigenvalues having negative real parts). Hence, the Center Manifold Theory (Carr 1981) can be used to analyze the dynamics of system (39). In particular, the Theorem of Castillo-Chavez and Song (2004), reproduced below for convenience, will be used to show that when ${\mathcal {N}}_{0}^{(1)} >1$, there exists a unique non trivial equilibrium point of system (39) (as shown in Lemma 4) which is locally asymptotically stable for ${\mathcal {N}}_{0}^{(1)}$ near 1.

Consider the case when ${\mathcal {N}}_{0}^{(1)}=1$. Suppose further, that $\alpha _1=\alpha _1^*$ is chosen as a bifurcation parameter solving ${\mathcal {N}}_{0}^{(1)}=1$ gives

$$\begin{aligned} \alpha _1=\alpha _1^*=\dfrac{\mu _2\mu _3}{r_2ea_{1}E_{s}^{0}-\mu _3}. \end{aligned}$$

(40)

Theorem 7

Castillo-Chavez and Song (2004) : Consider the following general system of ordinary differential equations with parameter $\phi$:

$$\begin{aligned} \dfrac{dz}{dt}=f(z,\phi ), \quad f: {\mathbb {R}}^{n}\times {\mathbb {R}} \rightarrow {\mathbb {R}} \quad and \quad f\in {\mathcal {C}}^{2}({\mathbb {R}}^{n},{\mathbb {R}}), \end{aligned}$$

(41)

where 0 is an equilibrium point of the system (that is, $f(0;\phi ) \equiv 0$ for all $\phi$) and assume

$A_1$::

$A=D_zf(0,0)=\frac{\partial f_i}{\partial z_j}(0,0)$ is the linearisation matrix of system (41) around the equilibrium 0 with a evaluated at 0. Zero is a simple eigenvalue of A and all other eigenvalues of A have negative real parts;

$A_2$::

Matrix A has a right eigenvector u and a left eigenvector v corresponding to the zero eigenvalue. Let $f_k$ be the $k^{th}$ component of f and

$$\begin{aligned} \begin{array}{lcl} {\tilde{a_1}}&{}=&{}\sum \limits _{k,i,j=1}^{n}v_{k}u_{i}u_{j}\frac{\partial ^{2}f_{k}}{\partial z_{i}\partial z_j}(0,0),\\ {\tilde{b_1}}&{}=&{}\sum \limits _{i,k=1}^{n} v_ku_i\frac{\partial ^{2}f_k}{\partial z_i \partial \phi }(0,0). \end{array} \end{aligned}$$

(42)

The local dynamics of system (41) around 0 are totally determined by $a_1$ and $b_1$

• (i) ${\tilde{a_1}>0}, {\tilde{b_1}>0}.$ When $\phi <0$ with $|\phi |\ll 1$, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium; when $0< \phi \ll 1$, 0 is unstable and there exists a negative and locally asymptotically stable equilibrium;

• (ii) ${\tilde{a_1}<0}, {\tilde{b_1}<0}$. When $\phi <0$ with $|\phi |\ll 1$, 0 is unstable; when $0<\phi \ll 1$, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium;

• (iii) ${\tilde{a_1}>0}, {\tilde{b_1}<0}$. When $\phi <0$ with $|\phi |\ll 1$, 0 is unstable, and there exists a locally asymptotically stable negative equilibrium; when $0 <\phi \ll 1$, 0 is stable, and a positive unstable equilibrium appears;

• (iv) ${\tilde{a_1}<0},{\tilde{b_1}>0}$. When $\phi$ changes from negative to positive, 0 changes its stability from stable to unstable. Correspondingly a negative unstable equilibrium becomes positive and locally asymptotically stable.

Eigenvectors of $J_{\alpha _{1}}$ for the cure when ${\mathcal {N}}_{0}^{(1)}=1$ it can be shown that the Jacobian $J_{\alpha _{1}}$ of system (39) at $\alpha _1=\alpha _1^*$ has a right eigenvector (corresponding to zero eigenvalues given by $u =(u_1, u_2, u_3)^{T}$ are given by

$$\begin{aligned} u_1=\dfrac{a_1E_{s}^{0}}{\mu _1-r_1}\left( 1+\dfrac{\alpha _1}{\mu _3}\right) u_2, \qquad u_2> 0 \quad \text {and} \quad u_3=\dfrac{\alpha _1}{\mu _3}u_2. \end{aligned}$$

(43)

Similarly, the left eigenvector of $J(Q_{0}^{(2)})$ (corresponding to the zero eigenvalue denoted by $v=(v_1, v_2, v_3)^{T}$ are given by

$$\begin{aligned} v_1=0, \qquad v_2=\dfrac{\alpha _1}{\alpha _1+\mu _2}v_3 \quad \text {and} \quad v_3>0. \end{aligned}$$

(44)

${{{\underline{\textit{\textbf{Computation of}}}}} \tilde{b_1}}:$ For system (39), the associated non-zero partial derivatives of f at $Q_{0}^{(2)}$ are given by

$$\begin{aligned} \dfrac{\partial ^{2}f_2}{\partial z_2 \partial \alpha _1}=-1 \quad \text {and} \quad \dfrac{\partial ^{2}f_3}{\partial z_2 \partial \alpha _1}=1. \end{aligned}$$

(45)

Then, it follows that

$$\begin{aligned} {\tilde{b_1}}&=\sum _{i,k=1}^{3} v_ku_i\frac{\partial ^{2}f_k}{\partial z_i \partial \alpha _1}(0,0),\\&=v_2u_2\dfrac{\partial ^{2}f_2}{\partial z_2 \partial \alpha _1}+v_3u_2\dfrac{\partial ^{2}f_3}{\partial z_2 \partial \alpha _1},\\&=\left( v_3-v_2\right) u_2,\\&=\left( v_3-\dfrac{\alpha _1}{\alpha _1+\mu _2}v_3\right) u_2,\\&=\left( 1-\dfrac{\alpha _1}{\alpha _1+\mu _2}\right) u_2v_3>0. \end{aligned}$$

${{{\underline{\varvec{Computation of:}}}} {\tilde{a_1}}}$ For system (39), the associated non-zero partial derivatives of f at $Q_{0}^{2}$ are given by

$$\begin{aligned} \dfrac{\partial ^{2}f_2}{\partial z_2 \partial z_3}=\dfrac{\partial ^{2}f_2}{\partial z_3\partial z_2}=-2\dfrac{r_2ea_{1}E_{s}^{0}}{K_2}\quad \text {and} \quad \dfrac{\partial ^{2}f_2}{\partial z_1\partial z_3}=\dfrac{\partial ^{2}f_2}{\partial z_3\partial z_1 }=ea_1r_2. \end{aligned}$$

(46)

Then, it follows that

$$\begin{aligned} {\tilde{a_1}}&=\sum _{k,i,j=1}^{3}v_{k}u_{i}u_{j}\frac{\partial ^{2}f_{k}}{\partial z_{i}\partial z_j}(0,0),\\&=v_2\sum _{i,j=1}^{3}u_{i}u_{j}\frac{\partial ^{2}f_{2}}{\partial z_{i}\partial z_j}(0,0),\\&=v_2u_1\sum _{j=1}^{3}u_{j}\frac{\partial ^{2}f_{2}}{\partial z_{1}\partial z_j}(0,0)+v_2u_2\sum _{j=1}^{3}u_{j}\frac{\partial ^{2}f_{2}}{\partial z_{2}\partial z_j}(0,0)+v_2u_3\sum _{j=1}^{3}u_{j}\frac{\partial ^{2}f_{2}}{\partial z_{3}\partial z_j}(0,0),\\&=2v_2\left( u_1u_3\dfrac{\partial ^{2}f_2}{\partial z_1\partial z_3}(0,0) + 2u_2u_3\dfrac{\partial ^{2}f_2}{\partial z_2\partial z_3}(0,0)\right) ,\\&=2ea_1r_2v_2u_3\left( \dfrac{a_1E_{s}^{0}}{\mu _1-r_1}\left( 1+\dfrac{\alpha _1}{\mu _3}\right) u_2 -2\dfrac{E_{s}^{0}}{K_2}u_2\right) ,\\&=-2ea_1r_2v_2u_3\left( \dfrac{a_1K_1}{r_1}\left( 1+\dfrac{\alpha _1}{\mu _3}\right) u_2 +2\dfrac{E_{s}^{0}}{K_2}u_2\right) ,\\&=-2ea_1r_2v_2u_{2}^{2}\dfrac{\alpha _1}{\mu _3}\left( \dfrac{a_1K_1}{r_1}\left( 1+\dfrac{\alpha _1}{\mu _3}\right) +2\dfrac{E_{s}^{0}}{K_2}\right) <0. \end{aligned}$$

Thus, ${\tilde{a_1}<}0$ and from Theorem 3 the non trivial equilibrium $Q^{*}$ of system (9) is locally asymptotically stable when ${\mathcal {N}}_{0}^{(1)}>1$. This achieves the proof. $\square$

Appendix D

Herein, we present a result used for the stability of the trivial equilibrium $Q_{0}^{(22)}$ of system (23).

Lemma 7

Let M be a square Metzler matrix written in block form

$$\begin{aligned} M=\left( \begin{array}{cc} A&{} B\\ C&{} D \end{array}\right) , \end{aligned}$$

where A and D are square matrices. Then, the matrix M is Metzler stable if and only if matrices. A and $D-CA^{-1}B$ (or D and $A-CD^{-1}B$) are Metzler stable (Aleksandar 2019; Kumpati and Robert 2009).

Appendix E: Proof of Theorem 5

We will prove this theorem using a result by Castillo-Chavez and Song in Castillo-Chavez and Song (2004). To do this, we do the following change of variables: $(z_1, z_2, z_3, z_4, z_5) = (E_s, P_v, P_M, P_s, V)$. In this case, system (23) can be written in the following compact form:

$$\begin{aligned} {\dot{z}}=f(z), \end{aligned}$$

where $f=(f_1, f_2, f_3, f_4, f_5)$ is defined as follows:

$$\begin{aligned} \left\{ \begin{array}{ll} \dot{z_1} ~=~ r_1\left( z_1+E_{s}^{0}\right) \left( 1-\dfrac{z_1+E_{s}^{0}}{K_1}\right) -a_1(z_2+z_3+z_4)\left( z_1+E_{s}^{0}\right) { \left( z_5+1\right) }- \mu _1\left( z_1+E_{s}^{0}\right) =f_1,\\ \dot{z_{2}} ~=~r_2ea_{1}\left( z_1+E_{s}^{0}\right) (z_3+z_4)\left( 1-\dfrac{z_2}{K_2}\right) -(\alpha _1+\mu _2)z_2-\delta (K-ea_1\left( z_1+E_{s}^{0}\right) )z_2=f_2,\\ \dot{z_{3}} ~=~ \alpha _1z_2-\mu _3z_3=f_3,\\ \dot{z_{4}} ~=~\delta (K-ea_1\left( z_1+E_{s}^{0}\right) )z_2-\left( \mu _4+\dfrac{\theta }{1+ea_1\left( z_1+E_{s}^{0}\right) z_4}\right) z_4=f_4,\\ \dot{z_{5}} ~=~\alpha z_5+\beta z_4=f_5. \end{array} \right. \end{aligned}$$

(47)

Consider the case when ${\mathcal {N}}_{0}^{(2)}=1$. Suppose further, that $\alpha _1=\alpha _1^*$ is chosen as a bifurcation parameter solving ${\mathcal {N}}_{0}^{(2)}=1$ gives

$$\begin{aligned} \alpha _1=\alpha _1^*=\dfrac{(\mu _4+\theta )\mu _3\left( \mu _2+\delta (K-ea_1E_{s}^{0})\right) -\delta r_2ea_{1}E_{s}^{0}\mu _3(K-ea_1E_{s}^{0})}{r_2ea_{1}E_{s}^{0}(\mu _4+\theta )-(\mu _4+\theta )\mu _3} \end{aligned}$$

(48)

The Jacobian of system (47) at $Q_{0}^{(22)}$ is

$J(Q_{0}^{(22)})=\left( \begin{array}{ccccccc} \mu _1-r_1&{} -a_1E_{s}^{0}&{} -a_1E_{s}^{0}&{} -a_1E_{s}^{0}&{} r_1E_{s}^{0} -\dfrac{r_1}{K_1}E_{s}^{0}\\ 0&{} -(\alpha _1+\delta (K-ea_1E_{s}^{0})+\mu _2)&{} r_2ea_{1}E_{s}^{0}&{} r_2ea_{1}E_{s}^{0}&{} 0\\ 0&{} \alpha _1&{} -\mu _3&{} 0&{} 0 \\ 0&{} \delta (K-ea_1E_{s}^{0})&{} 0&{} -(\mu _4+\theta )&{} 0\\ 0&{} 0&{} 0&{} \beta &{} -\alpha \\ \end{array}\right) ,$

Eigenvectors of $J_{a^{*}}$ for the cure when ${\mathcal {N}}_{0}^{(2)}=1$ it can be shown that the Jacobian $J_{a^{*}}$ of system (47) at $\alpha _1=\alpha _1^{*}$ has a right eigenvector (corresponding to zero eigenvalues given by $u=(u_1, u_2, u_3, u_4, u_5)^{T}$ are given by

$$\begin{aligned}{} & {} u_1=\dfrac{1}{\mu _1-r_1}\left[ a_1E_{s}^{0}+\dfrac{ea_1\alpha _1E_{s}^{0}}{\mu _3}+\dfrac{ea_1\delta E_{s}^{0}(K-ea_1E_{s}^{0})}{\mu _4+\theta }-\dfrac{\beta \delta (K-ea_1E_{s}^{0})}{\alpha (\mu _4+\theta )}\left( r_1E_{s}^{0}-\dfrac{r_1}{K_1}E_{s}^{0}\right) \right] u_2, \qquad u_2> 0, \end{aligned}$$

(49)

$$\begin{aligned}{} & {} u_3=\dfrac{\alpha _1}{\mu _3}u_2, \qquad u_4= \dfrac{\delta (K-ea_1E_{s}^{0})}{\mu _4+\theta }u_2 \quad \text {and} \quad u_5=\dfrac{\beta \delta (K-ea_1E_{s}^{0})}{\alpha (\mu _4+\theta )}u_2. \end{aligned}$$

(50)

Similarly, the left eigenvector of $J(Q_{0}^{(22)})$ (corresponding to the zero eigenvalue denoted by $v=(v_1, v_2, v_3, v_4, v_5)^{T}$ are given by

$$\begin{aligned} v_1=0, \qquad v_2>0,\qquad v_3=\dfrac{r_2ea_{1}E_{s}^{0}}{\mu _3}v_2, \qquad v_4=\dfrac{r_2ea_{1}E_{s}^{0}}{(\mu _4+\theta )}v_2 \quad \text {and} \quad v_5=0. \end{aligned}$$

(51)

${{{\underline{\varvec{Computation of}}}} {\tilde{b_1}}}$: For system (47), the associated non-zero partial derivatives of f at $Q_{02}$ are given by

$$\begin{aligned} \dfrac{\partial ^{2}f_2}{\partial z_2 \partial \alpha _1}=-1 \quad \text {and} \quad \dfrac{\partial ^{2}f_3}{\partial z_2 \partial \alpha _1}=1. \end{aligned}$$

(52)

Then, it follows that

$$\begin{aligned} {\tilde{b_1}}&=\sum _{i,k=1}^{5} v_ku_i\frac{\partial ^{2}f_k}{\partial z_i \partial \alpha _1}(0,0),\\&=v_2u_2\dfrac{\partial ^{2}f_2}{\partial z_2 \partial \alpha _1}+v_3u_2\dfrac{\partial ^{2}f_3}{\partial z_2 \partial \alpha _1},\\&=\left( v_3-v_2\right) u_2,\\&=\left( \dfrac{r_2ea_{1}E_{s}^{0}}{\mu _3}-1\right) u_2v_2>0. \end{aligned}$$

${{{\underline{\varvec{Computation of:}}}} {\tilde{a_1}}}$ For model system (47), the associated non-zero partial derivatives of f at $Q_{02}$ are given by

$$\begin{aligned} \dfrac{\partial ^{2}f_2}{\partial z_3\partial z_1}=\dfrac{\partial ^{2}f_2}{\partial z_1\partial z_3}=r_2ea_1,\ \dfrac{\partial ^{2}f_2}{\partial z_4\partial z_1}=\dfrac{\partial ^{2}f_2}{\partial z_1\partial z_4}=r_2ea_1, \ \dfrac{\partial ^{2}f_2}{\partial z_3\partial z_2}=\dfrac{\partial ^{2}f_2}{\partial z_2\partial z_3}=-\dfrac{r_2}{K_2}ea_1E_{s}^{0}, \end{aligned}$$

(53)

and

$$\begin{aligned} \dfrac{\partial ^{2}f_2}{\partial z_4\partial z_2}=\dfrac{\partial ^{2}f_2}{\partial z_2\partial z_4}=-\dfrac{r_2}{K_2}ea_1E_{s}^{0}, \ \dfrac{\partial ^{2}f_2}{\partial z_4\partial z_3}=\dfrac{\partial ^{2}f_2}{\partial z_3\partial z_4}=-\dfrac{r_2}{K_2}ea_1E_{s}^{0},\ \dfrac{\partial ^{2}f_4}{\partial z_2\partial z_1}=\dfrac{\partial ^{2}f_4}{\partial z_1\partial z_2}=-\delta ea_1, \ \dfrac{\partial ^{2}f_4}{\partial ^{2}z_4}=\theta . \end{aligned}$$

(54)

Then, it follows that

$$\begin{aligned} \begin{array}{lcl} {\tilde{a_1}}&{}=&{}\sum _{k,i,j=1}^{5}v_{k}u_{i}u_{j}\frac{\partial ^{2}f_{k}}{\partial z_{i}\partial z_j}(0,0),\\ &{}=&{}v_2\sum _{i,j=1}^{5}u_{i}u_{j}\frac{\partial ^{2}f_{2}}{\partial z_{i}\partial z_j}(0,0)+v_4\sum _{i,j=1}^{5}u_{i}u_{j}\frac{\partial ^{2}f_{4}}{\partial z_{i}\partial z_j}(0,0),\\ &{}=&{}v_2u_1\sum _{j=1}^{5}u_{j}\frac{\partial ^{2}f_{2}}{\partial z_{1}\partial z_j}(0,0)+v_2u_2\sum _{j=1}^{5}u_{j}\frac{\partial ^{2}f_{2}}{\partial z_{2}\partial z_j}(0,0)+v_2u_3\sum _{j=1}^{5}u_{j}\frac{\partial ^{2}f_{2}}{\partial z_{3}\partial z_j}(0,0)+v_2u_4\sum _{j=1}^{5}u_{j}\frac{\partial ^{2}f_{2}}{\partial z_{4}\partial z_j}(0,0),\\ &{}+&{}v_4u_1\sum _{j=1}^{5}u_{j}\frac{\partial ^{2}f_{4}}{\partial z_{1}\partial z_j}(0,0)+v_4u_2\sum _{j=1}^{5}u_{j}\frac{\partial ^{2}f_{4}}{\partial z_{2}\partial z_j}(0,0)+v_4u_4\sum _{j=1}^{5}u_{j}\frac{\partial ^{2}f_{4}}{\partial z_{4}\partial z_j}(0,0),\\ &{}=&{}2v_2\left( u_3u_1\dfrac{\partial ^{2}f_2}{\partial z_3\partial z_1}+u_4u_1\dfrac{\partial ^{2}f_2}{\partial z_4\partial z_1}+u_3u_2\dfrac{\partial ^{2}f_2}{\partial z_3\partial z_2}+u_4u_2\dfrac{\partial ^{2}f_2}{\partial z_4\partial z_2}+u_4u_3\dfrac{\partial ^{2}f_2}{\partial z_4\partial z_3}\right) \\ &{}+&{} 2v_4\left( u_2u_1\dfrac{\partial ^{2}f_4}{\partial z_1\partial z_2}+u_{4}^{2}\dfrac{\partial ^{2}f_4}{\partial z_4\partial z_4}\right) ,\\ &{}=&{}2v_2\left[ r_2ea_1u_1u_3+r_2ea_1u_1u_4-\dfrac{r_2}{K_2}ea_1E_{s}^{0}u_2u_3-\dfrac{r_2}{K_2}ea_1E_{s}^{0}u_4u_2-\dfrac{r_2}{K_2}ea_1E_{s}^{0}u_4u_3 \right] +2v_4\left[ -\delta ea_1u_1u_2+\theta u_{4}^{2}\right] . \end{array} \end{aligned}$$

(55)

Thus, depending on the values of the parameters of system (23), the value of ${\tilde{a_1}}$ can be positive or negative. So, since ${\tilde{b_1}}> 0$, the conclusion follows from Theorem (Castillo-Chavez and Song 2004) items (i) and (iv). $\square$