Global stability analysis of an age-structured model assessing the impact of Radopholus similis on banana-plantain production

TL;DR

This work develops an age-structured model of Radopholus similis infestation in banana–plantain crops, incorporating root-age dependent infection and two nematode compartments (free in soil $N_F$ and infesting within roots $N_I$). Using semigroup theory, it establishes existence, uniqueness, positivity, and boundedness of a mild solution, and under regularity assumptions upgrades to a classical solution; the disease-free state is analyzed via a threshold $ abla$ (written as $\\mathcal{N}$), which determines local stability. A semi-implicit Euler scheme is proposed and proven to be consistent, with numerical simulations illustrating disease-free and endemic scenarios and the impact of impulsive control strategies (nematicide applications) on yield. The results provide a rigorous framework for assessing nematode management in banana–plantain production, linking biologically interpretable parameters to stability and control outcomes and offering practical insights for integrated pest management.

Abstract

In this paper, we develop and analyse a mathematical model to investigate the interactions between banana and plantain plants and the nematode \textit{Radopholus similis}, a pest species occurring in banana plantations worldwide, with particularly high prevalence in Central Africa. The model incorporates root infection and mortality rates as functions of root age, providing a more realistic representation of the infection dynamics. We prove that the model is well-defined by establishing the existence and uniqueness of a mild solution using the theory of semi-groups for nonlinear evolutionary systems. We further show that the solution is positive and bounded. Under additional assumptions on the regularity of the parameters and the data, we prove that the mild solution is indeed a classical solution. We then study the asymptotic behaviour of the solution by deriving a threshold parameter \(\mathcal{N}\), which determines the stability of the disease-free equilibrium. Finally, we perform a numerical analysis of the proposed model using the semi-implicit Euler method. The biological consistency of the numerical solutions is established, and simulations are carried out to illustrate the theoretical results and estimate yield losses caused by nematodes. We conclude the numerical analysis by implementing an impulsive control strategy, which confirms that the use of nematicides - whether chemical or biological - helps to mitigate the devastating effects of nematodes and enhances crop yield.

Figures (7)

• Figure 1: Diagram of the nematode propagation in the banana-plantain plantation. State variables: Healthy plants (S), Infected plants (I), Free nematodes $(N_{F})$ and Infesting nematodes $(N_{I}).$ The logistic expressions of the model are given by: $(i):\, \beta(a)\mathcal{W}(P,N_{F}),$$(ii):\, \dfrac{d(a) N_{I}}{K_d+\int_a^{a_{\dagger}} I(a,t)da},$$(iii):\, e\int_{0}^{a_{\dagger}}\beta(a)\mathcal{W}(P,N_F)da,$$(iv):\,\alpha( \int_{0}^{a_{\dagger}}\beta(\cdot) S(\cdot\,,\,t)da + e \int_{0}^{a_{\dagger}}\beta(a)I(a,t))da) \mathcal{W}(P,N_{F})$ and $(v) :\, \rho N_{I}\dfrac{\int_0^{a_{\dagger}} d(a)I(a,t)da}{K_d+\int_0^{a_{\dagger}} I(a,t)da}\left(1-\dfrac{N_I}{K}\right).$
• Figure 2: Simulation in the case $0\leq \mathcal{N}<1$ of the dynamics of healthy plants $S(a,t)$, infected plants $I(a,t)$, free nematodes $N_F(t)$ and infesting nematodes $N_I(t)$ using the parameters given in Table \ref{['t1']} and equations \ref{['inc']}--\ref{['par']}.
• Figure 3: Simulation in the case $\mathcal{N} >1$ of the dynamics of healthy plants $S(a,t)$, infected plants $I(a,t)$, free nematodes $N_F(t)$ and Infesting nematodes $N_I(t)$ using the parameters given in Table \ref{['t1']} and equations \ref{['inc']}--\ref{['par']}.
• Figure 4: Simulation of the daily and final cumulative production without pests (gray curves) and with pests (blue curves), using the parameters given in Table \ref{['t1']} and equations \ref{['inc']}--\ref{['par']}.
• Figure 5: Simulation of the dynamics of healthy plants $S(a,t)$ and infected plants $I(a,t)$ under control, along with a comparison of the dynamics of free and infesting nematodes without control (blue curves) and with control (red curves), using the parameters provided in Table \ref{['t1']} and equations \ref{['inc']}-\ref{['par']}.

Theorems & Definitions (30)

• Remark 2.1
• Remark 3.1
• Proposition 3.1
• proof
• Proposition 3.2
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Proposition 3.3
• Definition 3.1