# Modelling of biological control strategies in a self-financing plantation

Aurelien Kambeu Youmbi, Suzanne Touzeau, Frédéric Grognard, Berge Tsanou

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Aurelien Kambeu Youmbi, Suzanne Touzeau, Frédéric Grognard, Berge Tsanou. Modelling of biological control strategies in a self-financing plantation. MPDEE 2025 - Conference on Mathematical Population Dynamics, Ecology and Evolution, May 2025, Bilbao, Spain. . hal-05166981

HAL Id: hal-05166981

https://inria.hal.science/hal-05166981v1

Submitted on 17 Jul 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# MODELLING OF BIOLOGICAL CONTROL STRATEGIES IN A SELF-FINANCING PLANTATION

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RÉPUBLIQUE
FRANÇAISE
Liberté
Égalité
Fraternité

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INRAE

Aurelien Kambeu Youmbi<sup>a,b,c</sup>, Suzanne Touzeau<sup>b,c</sup>, Frédéric Grognard<sup>c</sup>,
Berge Tsanou<sup>a</sup>

<sup>a</sup>University of Dschang, URMA, Cameroon

<sup>b</sup>Université Côte d'Azur, INRAE, CNRS, ISA, France

<sup>c</sup>Université Côte d'Azur, Inria, INRAE, CNRS, MACBES, France

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Scan Me!

## 1. RESEARCH QUESTIONS

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## 2. MODEL FORMULATION

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◆ Initial release: black-blue

◆ Continuous release: black-blue-red

◆ Periodic releases: black-gray (dashed lines)

$$ \dot{B}_y = \underbrace{\frac{r_1 k_B M}{1 + \eta M}}_{\text{new } B_y} + \underbrace{r_y B_y}_{\text{net growth}} - \underbrace{\psi_L L B_y}_{\text{consumption}} - \underbrace{\gamma B_y}_{\text{aging}} $$

$$ \dot{B}_a = \underbrace{\gamma B_y}_{\text{new } B_a} + \underbrace{r_a B_a}_{\text{net growth}} - \underbrace{h B_a}_{\text{harvest}} $$

$$ \dot{L} = \underbrace{c_L \psi_L L B_y}_{\text{growth}} - \underbrace{(\mu_L L^2 + \delta_L L)}_{\text{mortality}} - \underbrace{\frac{\psi_A L A}{1 + \alpha L + \beta A}}_{\text{predation}} $$

$$ \dot{A} = \underbrace{\lambda k_A M}_{\text{c. release}} + \underbrace{\frac{c_A \psi_A L A}{1 + \alpha L + \beta A}}_{\text{growth}} - \underbrace{\delta_A A}_{\text{mortality}} $$

$$ \dot{M} = \underbrace{q h B_a}_{\text{sale}} - \underbrace{k M}_{\text{withdrawal}} $$

$$ \dot{T} = \underbrace{k M}_{\text{new income}} - \frac{k_B M}{1 + \eta M} - k_A M $$

### ◆ Discrete part

$$ B_y(j\tau^+) = \phi_B B_y(j\tau) + \frac{r_1 k_B^d (M(j\tau) + qB_a(j\tau))}{1 + \eta^d (M(j\tau) + qB_a(j\tau))} $$

$$ B_a(j\tau^+) = 0 $$

$$ L(j\tau^+) = \phi_L L(j\tau) $$

$$ A(j\tau^+) = \phi_A A(j\tau) + \lambda k_A^d (M(j\tau) + qB_a(j\tau)) $$

$$ M(j\tau^+) = M(j\tau) + qB_a(j\tau) - k^d (M(j\tau) + qB_a(j\tau)) $$

$$ T(j\tau) = k^d (M(j\tau) + qB_a(j\tau)) $$

$$ - \frac{k_B^d (M(j\tau) + qB_a(j\tau))}{1 + \eta^d (M(j\tau) + qB_a(j\tau))} - k_A^d (M(j\tau) + qB_a(j\tau)) $$

## 3. MATHEMATICAL ANALYSIS AND SIMULATIONS

### Initial release

<table><thead><tr><th>Equilibria</th><th>Existence</th><th>Local stability</th></tr></thead><tbody><tr><td>Extinction equilibrium E<sub>0</sub></td><td>Always exists</td><td>R<sub>B</sub> ≤ 1</td></tr><tr><td>Pest-free equilibrium PFE<sup>e</sup></td><td>R<sub>B</sub> > 1</td><td>R<sub>L</sub> ≤ 1</td></tr><tr><td>Parasitoid-free equilibrium E<sup>*</sup></td><td>R<sub>L</sub> > 1</td><td>R<sub>A</sub> < 1</td></tr><tr><td>Coexistence equilibrium E<sup>**</sup></td><td>R<sub>A</sub> > 1</td><td>Yes or/and LC</td></tr></tbody></table>

A. 3 transcritical and 1 Hopf bifurcations

B. blue: LAS E<sup>**</sup>, red: stable limit cycle (LC)

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### Continuous release

<table><thead><tr><th>Equilibria</th><th>Existence</th><th>Local stability</th></tr></thead><tbody><tr><td>Extinction equilibrium E<sub>0</sub></td><td>Always exists</td><td>R<sub>B</sub> ≤ 1</td></tr><tr><td>Pest-free equilibrium PFE<sup>cr</sup></td><td>R<sub>B</sub> > 1</td><td>R<sub>L</sub><sup>c</sup> < 1</td></tr><tr><td>Coexistence equilibrium E<sup>c*</sup></td><td>R<sub>L</sub><sup>c</sup> > 1</td><td>Yes or/and LC</td></tr></tbody></table>

A. green: stable PFE<sup>cr</sup>, blue: stable E<sup>c*</sup>, red: limit cycle (LC)

B. R<sub>L</sub><sup>c</sup> = 0.99 < 1 (control cost: $250)

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### Periodic releases

<table><thead><tr><th>Solution</th><th>Existence</th><th>Local stability</th></tr></thead><tbody><tr><td>Extinction</td><td>Always exists</td><td>R<sub>B</sub><sup>d</sup> < 1</td></tr><tr><td>Periodic pest-free</td><td>R<sub>B</sub><sup>d</sup> > 1</td><td>R<sub>L</sub><sup>d</sup> < 1</td></tr><tr><td>Coexistence</td><td>R<sub>L</sub><sup>d</sup> > 1</td><td>chaos</td></tr></tbody></table>

A. R<sub>B</sub><sup>d</sup> = 0.99 < 1 (control cost $500)

B. R<sub>B</sub><sup>d</sup> = 1.11 > 1 (control cost $250)

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## 4. CONCLUSION

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Parasitoid establishment or control:

* possible with reasonable mutual interference $\beta$

* often leads to oscillations that can be dampened by continuous release

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Continuous release: best theoretical choice but not practical → more frequent releases?