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Study of the Population Dynamics of Busseola fusca, Maize Pest

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Abstract

Busseola fusca is a maize and sorghum pest that can cause significant damage to both crops. Given that maize is one of the main cereals grown in the worldwide, this pest is a major challenge for maize production and therefore for the economies of several countries . In this paper , based on the life cycle of B. fusca, we propose a mathematical model to study the population dynamics of this insect pest . A sensitivity analysis using the eFast method was performed to show the most important parameters of the model. We present the theoretical analysis of the model. More precisely, we derive a threshold parameter \({\mathcal {N}}_0\), called basic offspring number and show that the trivial equilibrium is globally asymptotically stable whenever \({\mathcal {N}}_0\le 1\), while if \({\mathcal {N}}_0>1\), the non trivial equilibrium is globally asymptotically stable. The theoretical results are supported by numerical simulations.

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Acknowledgements

The authors are grateful to the anonymous reviewers, and the Handling Editor, for their suggestions that have greatly improved the paper.

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Correspondence to Samuel Bowong.

Appendix: Cooperative Systems

Appendix: Cooperative Systems

In this appendix, we recalled some important results about cooperative systems.

Consider a system of ordinary differential equations:

$$\dot{x}=f(x),$$
(14)

where \(\dot{x}=\frac{dx}{dt},~~f:I\rightarrow {\mathbb {R}}^n,~~\text {and}~I\subseteq {\mathbb {R}}^n.\) We assume that the solution \(x(t,x_0)\) of system (14) exists and is unique with the initial conditions \(x(t_0)=x_0\). In other word, \(x(t,x_0)\) is the trajectory of the system (14) from the initial condition \(x_0.\)

In population dynamics, a system is cooperative if the increase in the population j favors the increase in the population i. The speed of variation of a state variable is an increasing function of the other state variables.

Definition A.1

The system (14) is said to be cooperative if for every \(i,j\in \{1,2,\ldots ,n\}\) such that \(i\ne j\), the function \(f_i(x_1,x_2,\ldots ,x_n)\) is monotone increasing with respect to \(x_j\)

If f is differentiable, the system (14) is cooperative if the Jacobian \(\frac{\partial f}{\partial x}\) is Metzler matrix, that is, if the non-diagonal elements of the Jacobian matrix are positive.

Theorem A.1

Let the system (14) be cooperative. Then, for every \(x_1, x_2\in I, ~x_1\le x_2\Rightarrow x(t,x_1)\le x(t,x_2),~t\in [0, \min \{T_{x_1},T_{x_2}\}[.\)

Note that \([0, T_{x_1}[\) is the maximal (non negative) interval of existence of \(x(t,x_1).\) The same for \([0, T_{x_2}[.\)

Theorem A.2

Consider the autonomous cooperative system \(\dot{x}=f(x)\). This system is positive if and only if \(f(0)\ge 0.\)

Theorem A.3

Let \(x_1,x_2\in I\) be such that \(x_1\le x_2,~[x_1,x_2]\in I\) and \(f(x_2)\le 0\le f(x_1).\) Then the system (14) defines a positive dynamical system on \([x_1, x_2].\) Moreover, if \([x_1,x_2]\) contains a unique equilibrium point, then this point is globally asymptotically stable on \([x_1,x_2].\)

For more information, the proof of the previous theorems can be found in (Farina and Rinaldi 2000; Anguelov et al. 2012; Smith 1995; Mailleret 2004).

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Ntahomvukiye, J.P., Temgoua, A. & Bowong, S. Study of the Population Dynamics of Busseola fusca, Maize Pest. Acta Biotheor 66, 379–397 (2018). https://doi.org/10.1007/s10441-018-9335-x

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