Understanding the mechanisms of locust growth can be very important in bringing these insects to the brink of extinction. For a mathematical description of their evolution from a global point of view, we can use a so-called "diffusion-convection" equation. Diffusion-convection equations are powerful tools for modeling certain aspects of locust population dynamics, particularly with regard to their dispersal and migration over large scales. However, to fully capture the complexity of locust behavior, including swarm formation and phase transitions between solitary and gregarious states, it is often necessary to combine them with other types of model. This takes advantage of the ability of diffusion-convection equations to model global trends while incorporating finer dynamics and complex interactions.

In this manuscript, we investigate the existence and uniqueness of a solution for a four-stage age-structured population dynamics model with spatial diffusion for locusts.

**Experimental setup: Biological control combined with physical barriers**. This makes it possible to reduce the locust population in a given area using a combination of biological control (biopesticides) and physical techniques (mechanical barriers).

Here’s a simple and effective conceptual diagram to illustrate the experimental process.

= [rectangle, rounded corners, minimum width=3cm, minimum height=1cm, text centered, draw=black, fill=blue!20] = [thick,->,>=stealth]

We now consider the following locust population dynamics model, which is a system of four equations. \[\begin{cases}
\partial_ty_1(b,t,x)+\partial_b \Big [v_1(b,T(t),x)y_1(b,t,x)\Big ]
-\beta \partial_b{^2} y_1(b,t,x) -\mu_{1} \Delta y_1(b,t,x)
\\
=-\Big [\theta_1(b,T(t), \textit{x})+\gamma_1(b,T(t), \textit{x}) \Big ] y_1(b,t,x), \ \ \ (b,t,x)\in \alpha_1,
\\
\\[0.5cm]
\partial_ty_2(b,t,x)+\partial_b \Big [
v_2(b,T(t),x)y_2(b,t,x)\Big ]
-\beta \partial_b{^2} y_2(b,t,x)-\mu_{2} \Delta y_2(b,t,x)
\\
=- \Big [\theta_2(b,T(t), \textit{x})+\gamma_2(b,T(t), \textit{x}) \Big ] y_2(b,t,x), \ \ \ (b,t,x)\in \alpha_2,
\\
\\[0.5cm]
\partial_ty_3(b,t,x)+\partial_b\Big [ v_3(b,T(t),x)y_3(b,t,x) \Big ]
-\beta \partial_b{^2} y_3(b,t,x)-\mu_{3} \Delta y_3(b,t,x)
\\
=-\theta_3(b,T(t), \textit{x}) y_3(b,t,x), \ \ \ (b,t,x)\in \alpha_3,
\\
\\[0.5cm]
\partial_ty_4(b,t,x)+\partial_b \Big [ v_4(b,T(t),x)y_4(b,t,x) \Big ]
-\beta \partial_b{^2} y_4(b,t,x) -\mu_{4} \Delta y_4(b,t,x)
\\
=-\theta_4(b,T(t), \textit{x}) y_4(b,t,x), \ \ \ (b,t,x)\in \alpha_4.
\label{1.1}
\end{cases}\]

Where: \(y_j(b,t, \textit{x}), \ \ 1\leq j \leq 4\), respectively represent the density of eggs, larvae, females and males of age \(b\) at time \(t\) in a geographical position \(x\) in the locust domain.

\(\theta_j(b,T(t), \textit{x}), \ \ 1\leq j \leq 4\), represents respectively the mortality rate of eggs, larvae, females and males of age b at temperature \(T(t)\) in a geographical position \(x\) in the locust domain.

\(\gamma_j(b,T(t), x), \ 1\leq j \leq 2\) represents the transition function of stage \(j\); it is in fact the rate of individuals of age \(b\) at temperature \(T(t)\) of stage \(j\) which pass to stage \(j + 1\) in a geographical position \(x\) of the domain and \(\gamma_3(b,T(t), \textit{x})\), is the egg-laying rate.

\(v_j(b,T(t), \textit{x}), \ \ 1\leq j \leq 4\), represents the growth rate of individuals of age \(b\) at temperature \(T(t)\) (at time \(t\)) of stage \(j\) in a geographical position \(x\) of the domain.

\(B_j , \ \ 1\leq j \leq 4\), represents the maximum age of individuals in stage \(j\), \(\beta\), \(\mu_j\) and \(\lambda\) (sex ratio) are constants and \(t\geqslant 0\).

The diffusion coefficient \(\mu_j\) represents the effects of dispersion of individuals or the different movements of individuals in space.

The diffusion coefficient \(\beta\) represents the effects of dispersion or the various modifications in the organism of individuals during their development, \(\alpha_j =(0,B_j)\times (0,+\infty )\times \Omega, \ \ 1\leq j \leq 4\).

The boundary conditions and initial conditions are then expressed as follows \[\begin{cases}
\Big [v_1{(b,T(t),x)} y_1(b,t, x) -\beta \partial_b y_1(b,t,x) \Big ]_{b=0} = \displaystyle{\int_{0}^{B_3}\gamma_3(b,T(t),x) y_3(b,t,x) db},
\\[0.5cm]
\Big [v_2{(b,T(t),x)} y_2(b,t, x) -\beta \partial_b y_2(b,t,x) \Big ]_{b=0} =\displaystyle{ \int_{0}^{B_1}\gamma_1(b,T(t),x) y_1(b,t,x) db},
\\[0.5cm]
\Big [v_3{(b,T(t),x)} y_3(b,t, x) -\beta \partial_b y_3(b,t,x) \Big ]_{b=0} = \displaystyle{\int_{0}^{B_2} \lambda \gamma_2(b,T(t),x) y_2(b,t,x) db},
\\[0.5cm]
\Big [v_4{(b,T(t),x)} y_4(b,t, x) -\beta \partial_b y_4(b,t,x)\Big ]_{b=0} = \displaystyle{\int_{0}^{B_2}(1- \lambda) \gamma_2(b,T(t),x) y_2(b,t,x) db},
\\[0.5cm]
\partial_\eta y_j(b,t,x) = 0, \quad \quad \quad (b,t,x) \in \ (0,B_j) \times (0,+\infty) \times \partial\Omega ,
\\[0.5cm]
y_j(b,0,x) = y_j{^0}(b,x), \quad (b,x)\in \ (0,B_j) \times \Omega , \quad 1\leq j \leq 4 ,
\\[0.5cm]
y_j(B_j,t,x) = 0, \quad \quad (t,x) \in \ (0,+\infty) \times \Omega, \quad 1\leq j \leq 4, \label{1.2}
\\[0.5cm]
\end{cases}\] with \(t\geqslant 0\).

Here the following expression \(\displaystyle{\int_{0}^{B_j} \lambda_j \gamma_j(b,T(t),x) y_j(b,t,x) \ db},\) where \(1\leq j \leq 3\), designates respectively the distribution of larvae, females and eggs newborn at time \(t\) in a geographical position \(x\). With \((\lambda_1, \lambda_2, \lambda_3 ) = (1, \lambda, 1)\).

We also have the following expression \(\displaystyle{\int_{0}^{B_2}(1- \lambda) \gamma_2(b,T(t),x) y_2(b,t,x) \ db},\) which designates

the distribution of male newborns, where \(\lambda\) is the sex ratio.

Example of a diagram describing the stages and passage of locusts.

Thus, for the mathematical analysis of our model, we make the following assumptions about the functions of the locust model.

\(H_1\) : Growth functions \(v_j{(b,T(t),x)}\) are strictly positive and bounded, \[0< \hbox{min} \ v_j < v_j(b,T(t),x) < \hbox{max} \ v_j, \quad \forall (b,t,x) \in \alpha_j , \quad 1 \leq j \leq 4. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\] We could also remember that \(v_j(b,T(t),x)\) is continuously derivable with respect to the age variable, indeed \(v_j(b,T(t),x) \in C^1([0,B_j])\). Then there are positive constants \(A_j\) such that :

\[\left \Vert \frac{\partial v_j(b,T(t),x)}{\partial b} \right \Vert_{\infty} \leq A_j , \
\forall (b,t,x) \in \alpha_j , \quad 1 \leq j \leq 4.\] \(H_2\) : The transition functions \(\quad \gamma_j(b,T(t), \textit{x}), \quad 1\leq j \leq 4\) are also positive and bounded with respect to the age variable.

\(H_3\) : The functions of mortality \(\theta_j(b,T(t), \textit{x}), 1\leq j \leq 4\), \(\theta_j(b,T(t), \textit{x}) \in L^{\infty}((0,B_j) \times (0,T) \times \Omega )\) and are positive as well.

\(H_4\) : The initial conditions \(\quad y_j{^0}(b,0,x)
\in L^{2}((0,B_j) \times \Omega )\) and are positive.

\(H_5\) : The temperature function \(T(t)\) is measurable.

In this section we assume \(Q_j = (0,B_j) \times (0,T) \times \Omega ,\) with \(1\leq j \leq 4\); where \(T\) is a strictly positive real number.

Let \(\sigma\) be a positive constant and adopt the following change of variable \(y_j(b,t,x) = e^{\sigma t}\hat{y}_j(b,t,x) ,\)

\(1\leq j \leq 4\), in the system ([1.1]) and ([1.2]) which we’ll be using next.

Before stating one of the results coming from our article, we’ll start by giving the following definition:

[2.86] For all \(T >0, \ (\hat{y}_1, \hat{y}_2, \hat{y}_3, \hat{y}_4)\) is a solution of ([1.1])-([1.2]) if \((\hat{y}_1, \hat{y}_2, \hat{y}_3, \hat{y}_4)\) check: \[\begin{cases} \hat{y}_k \in L^2((0,T);\Gamma), \ \partial_t \hat{y}_k \in L^2((0,T); H^{-1}((0,B) \times \Omega )), \\[0.5cm] \displaystyle{ \int_{0}^{B} \int_{\Omega}\Big [ \partial_t \hat{y}_k (b,t,x) \ \varphi(b,x) -[ \lambda_i\gamma_i (b,T(t),x)\ \hat{y}_k (b,t,x) \varphi (0,x)] - [v_k (b,T(t),x) \hat{y}_k (b,t,x) \partial_b \varphi (b,x) ] } \\[0.5cm] + [\beta \partial_b \hat{y}_k (b,t,x) \partial_b \varphi (b,x) ] + \ \ \mu_{k} \nabla \hat{y}_k (b,t,x). \nabla \varphi(b,x) + [\theta_k +\gamma_k + \sigma]\hat{y}_k (b,t,x) \ \varphi(b,x) \Big ] \ dx \ db = 0, \\ \ for \ all \ \ \varphi \in \Gamma = \lbrace g \in H^1((0,B) \times \Omega)\vert \ g(B,x) = 0 \rbrace ,\ a.e. \ t \in \ (0,T), \ \ \\[0.5cm] \partial_\eta \hat{y}_k (b,t,x) = 0, \quad a.e. \ (b,t,x) \in \ (0,B) \times (0,T) \times \partial\Omega, \\[0.5cm] \hat{y}_k (b,0,x)= \hat{y}_k^{0}(b,x), \quad a.e. \ (b,x) \in \ (0,B) \times \Omega , \end{cases}\] where \((k,i,\lambda_i)\in \left\lbrace (1,3,1),(2,1,1),(3,2,\lambda),(4,2,1-\lambda)\right\rbrace\).

We are now able to present the main results.

[2.84] Assume that conditions \((H_1)-(H_5)\) are satisfied and that the following conditions \[\hbox{min} \ v_j \geq \lambda_i \sqrt{B}_j \Vert \gamma_j \Vert_{L^{\infty}((0,B)\times\Omega)} \ \ \ \hbox{with} \ \ (j,i,\lambda_i) \in \Big \lbrace (1,3,1), (2,1,1),(3,2,\lambda ) \Big \rbrace,\] are also satisfied. Hence the system ([1.1])-([1.2]) has a unique positive local solution.

So, to prove the existence, uniqueness, regularity and general properties of our problems ([1.1]) and ([1.2]), we’ll use Galerkin’s variational method and Banach’s fixed point theorem. For this we refer you to some references of our article such as: ,,, et

By the variational method, system ([1.1])-([1.2]) has a solution in the sense of Definition ([2.86])

This result is given by the following theorem:

[2.85] Let \(y_j{^0} \in L^2((0,B) \times\Omega)\) and \(T > 0\). The space \(L^2((0,B) \times\Omega)\) identifies with its dual.

let us assume that: \[P_j(t) = \int_{0}^{B_j} \int_{\Omega} \lambda_j \gamma_j \ \overline{y}_j \ dx \ db, \ with \ (j,\lambda_j)\in \left\lbrace (1,1),(2,\lambda),(3,1)\right\rbrace\]

and suppose that \(\ P_j \in L^2((0,T))\) , \(1 \leq j \leq 3\).

So, the problem([1.1])-([1.2]) has a unique solution satisfying the following system: \[\begin{cases}
\hat{y}_j \in L^2((0,T);\Gamma), \ \partial_t \hat{y}_j \in L^2((0,T); H^{-1}((0,B) \times \Omega )),
\\[0.5cm]
\displaystyle{ \int_{0}^{B} \int_{\Omega} \Big[ \partial_t \hat{y}_j (b,t,x)\varphi(b,x)- [v_j (b,T(t),x) \hat{y}_j (b,t,x) \partial_b \varphi (b,x)]}
+ [ \beta \partial_b \hat{y}_j (b,t,x) \partial_b \varphi (b,x) ]
\\[0.5cm]
+ \ \mu_j \nabla \hat{y}_j (b,t,x). \nabla \varphi (b,x)
+ [\theta_j +\gamma_j + \sigma_0]\hat{y}_j (b,t,x) \ \varphi(b,x) \Big] \ dx \ db \
- \ P_j(t) \varphi (0,x) = 0,
\\[0.5cm]
\ for \ all \ \varphi \in \Gamma= \lbrace g \in H^1((0,B) \times \Omega)\vert \ g(B,x) = 0 \rbrace ,\
\\[0.5cm]
\partial_\eta \hat{y}_j (b,t,x) = 0, \quad a.e. \ (b,t,x) \in \ (0,B) \times (0,T) \times \partial\Omega,
\\[0.5cm]
\hat{y}_j (b,0,x)= \hat{y}_j^{0}(b,x), \quad a.e. \ (b,x) \in \ (0,B) \times \Omega. \label{2.9}
\end{cases}\]

Moreover, if \(\ \hbox{max} \ v_j \geq 1\), we have the following estimate: \[\left \Vert \hat{y}_j(.,t,.) \right \Vert^{2}_ {L^2((0,B) \times\Omega)} \ + \ \beta \int_{0}^{t}
\left \Vert \nabla_b \hat{y}_j(.,s,.) \right \Vert^{2}_ {L^2( (0,B) \times \Omega )} \ ds + \mu \int_{0}^{t}
\left \Vert \nabla_x \hat{y}_j(.,s,.) \right \Vert^{2}_ {L^2( \Omega \times (0,B) ) } \ ds\] \[+ \int_{0}^{t}
\left \Vert \hat{y}_j(.,s,.) \right \Vert^{2}_ {L^2( (0,B) \times \Omega ) } \ ds \leq \ \left \Vert \hat{y}^{0}_j\right \Vert^{2}_ {L^2((0,B)\times\Omega)} + \ \ \left \Vert P_j \right \Vert^{2}_{L^2((0,T))}, \ \ \quad \hbox{for} \quad 1 \leq j \leq 3 \label{2.4}\]

We first demonstrate the inequality of the previous estimate ([2.4]), which in fact reassures us of the uniqueness of the solution

Also, with the proof of the theorem ([2.84]) we obtain a positive definite matrix, which shows that our system admits a unique solution.

Still in the proof of the theorem we have the following lemma:

[2.87] Let \(\Theta\) be defined by: \[\Theta \ : \ \ \prod_{j=1}^{3} {L^2(Q_j )^{+}} \ \ \longrightarrow \ \ \prod_{j=1}^{3} {L^2(Q_j )^{+}}\] \[( \overline{y}_1, \overline{y}_2, \overline{y}_3 ) \ \ \longmapsto \ \ ( \hat{y}_1, \hat{y}_2, \hat{y}_3 )\]

where

\[{L^2(Q_j )^{+}} = \Big \lbrace \varphi \in {L^2(Q_j )} \vert \varphi \geq 0 \Big \rbrace\]

Assume the hypotheses \((H_1)-(H_5)\) and assume that \[min \ v_j \geq \lambda_i \sqrt{B}_j \Vert \gamma_j \Vert_{L^{\infty}((0,B)\times\Omega)} \label{3.7}\] with \((j,i,\lambda_i) \in \Big \lbrace (1,3,1), (2,1,1),(3,2,\lambda ) \Big \rbrace\). Then \(\Theta\) is a contraction.

With the proof of the lemma ([2.87]), we arrive at the inequality ([3.18]) below. \[\left \Vert (\hat{y}_1, \ \hat{y}_2, \ \hat{y}_3) \right \Vert_Z \leq \frac{K}{L} \left \Vert(\overline{y}_1, \ \overline{y}_2, \ \overline{y}_3 )\right \Vert_Z.\label{3.18}\] Hence by choosing the parameter \(\sigma > 0\) sufficiently large in the inequality ([3.18]) we obtain \[\frac{K}{L} < 1.\] So \(\Theta\) is a contraction of \(Z\) in \(Z\).

From the lemma ([2.87]), \(\Theta\) is a contraction.

We can now show the proof of the theorem ([2.84])

Knowing finally from the lemma ([2.87]) that \(\Theta\) is a contraction. Then \(\Theta\) has a unique fixed point. Therefore, there is a unique triplet \[( \overline{y}_1, \overline{y}_2, \overline{y}_3 ) \in \prod_{j=1}^{3} {L^2(Q_j )^{+}} \ \ \hbox{such that} \ \ (\hat{y}_1, \ \hat{y}_2, \ \hat{y}_3) = \Theta ( \overline{y}_1, \overline{y}_2, \overline{y}_3 ) = ( \overline{y}_1, \overline{y}_2, \overline{y}_3 )\] and solution of ([1.1])-([1.2]).

\[\hbox{Then, from the previous calculations, we obtain the existence and uniqueness of a positive} \ \ \ \ \ \] \[\hbox{local solution of the system (\ref{1.1}})-{(\ref{1.2})} \ \hbox{on} \ \prod_{j=1}^{3} {L^2(Q_j )^{+}} . \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \] \[\hbox{ Finally we obtain the existence and the uniqueness of a local positive solution of (\ref{1.1})-(\ref{1.2}) on } \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\] \[\prod_{j=1}^{4} {L^2(Q_j )^{+}}. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \]

REFERENCES