We develop an eco-epidemiological model consisting of susceptible plant biomass, infected plant biomass, and pest density. The pest density plays a dual role by feeding on plant biomass and contributing to effective vector-mediated disease transmission. Allomone effects are included to reduce the net contribution of susceptible and infected plant consumption to pests' growth. The model accounts for direct plant-to-plant transmission, vector-mediated transmission, herbivory, and chemical plant defense. We analyze the existence and uniqueness, non-negativity, boundedness, equilibrium points, invasion thresholds, and local stability. The Li–Muldowney geometric approach is used to study the global stability of the coexistence equilibrium. A transcritical bifurcation characterizes the threshold transition in the direct transmission dynamics on the pest-free invariant subspace, while a codimension-two threshold interaction occurs when the direct transmission and pest density invasion thresholds are simultaneously satisfied. A Hopf bifurcation indicates the onset of oscillatory coexistence, and numerical simulations further show how vector-mediated transmission can amplify long-term fluctuations. The Latin hypercube sampling (LHS) and partial rank correlation coefficients (PRCCs) sensitivity results identify direct transmission, infected-plant removal, carrying capacity, and pest mortality as important parameters associated with the long-term mean state variables, with pest mortality showing the strongest association with mean pest density. The simulations also show that allomone-mediated pest suppression can reduce pest density and stabilize coexistence. An optimal control problem with treatment and insecticide interventions is then formulated and characterized through Pontryagin's minimum principle. For the selected objective function weights and control bounds, the combined treatment–insecticide strategy provides the strongest reduction in infected plant biomass and pest density among the scenarios considered, while maintaining susceptible plant biomass near the carrying capacity.
Citation: Taufan Talib, Agus Suryanto, Isnani Darti, Wuryansari Muharini Kusumawinahyu. Dynamic analysis and optimal control of a plant–pest model with dual transmission and allomone effects[J]. Mathematical Biosciences and Engineering, 2026, 23(11): 3273-3319. doi: 10.3934/mbe.2026113
We develop an eco-epidemiological model consisting of susceptible plant biomass, infected plant biomass, and pest density. The pest density plays a dual role by feeding on plant biomass and contributing to effective vector-mediated disease transmission. Allomone effects are included to reduce the net contribution of susceptible and infected plant consumption to pests' growth. The model accounts for direct plant-to-plant transmission, vector-mediated transmission, herbivory, and chemical plant defense. We analyze the existence and uniqueness, non-negativity, boundedness, equilibrium points, invasion thresholds, and local stability. The Li–Muldowney geometric approach is used to study the global stability of the coexistence equilibrium. A transcritical bifurcation characterizes the threshold transition in the direct transmission dynamics on the pest-free invariant subspace, while a codimension-two threshold interaction occurs when the direct transmission and pest density invasion thresholds are simultaneously satisfied. A Hopf bifurcation indicates the onset of oscillatory coexistence, and numerical simulations further show how vector-mediated transmission can amplify long-term fluctuations. The Latin hypercube sampling (LHS) and partial rank correlation coefficients (PRCCs) sensitivity results identify direct transmission, infected-plant removal, carrying capacity, and pest mortality as important parameters associated with the long-term mean state variables, with pest mortality showing the strongest association with mean pest density. The simulations also show that allomone-mediated pest suppression can reduce pest density and stabilize coexistence. An optimal control problem with treatment and insecticide interventions is then formulated and characterized through Pontryagin's minimum principle. For the selected objective function weights and control bounds, the combined treatment–insecticide strategy provides the strongest reduction in infected plant biomass and pest density among the scenarios considered, while maintaining susceptible plant biomass near the carrying capacity.
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Figures(15) / Tables(5)
Taufan Talib, Agus Suryanto, Isnani Darti, Wuryansari Muharini Kusumawinahyu. Dynamic analysis and optimal control of a plant–pest model with dual transmission and allomone effects[J]. Mathematical Biosciences and Engineering, 2026, 23(11): 3273-3319. doi: 10.3934/mbe.2026113
Schematic diagram of the plant–pest model with direct transmission, effective vector-mediated transmission, herbivory, and allomone-mediated suppression. The pest population contributes to plant damage through herbivory and to disease transmission through an effective infectious vector pressure. Allomone effects reduce the net demographic gain obtained by pests from plant consumption
Time series of System (2.1) for several positive initial conditions, illustrating numerical convergence toward the coexistence equilibrium $ E_3 $
PRCC values of the model parameters for the time-averaged susceptible plant biomass, infected plant biomass, and pest density over the interval $ 600\leq t\leq1200 $. The dashed lines at $ |\mathrm{PRCC}| = 0.4 $ indicate relatively strong associations and do not represent a statistical significance threshold
Biological feasibility and local stability of coexistence equilibria under two parameter sets. (a) Feasibility profiles $ I^*(S) $ and $ V^*(S) $, together with the infected plant positivity thresholds $ S = d_V/c_S $. Filled circles indicate biologically feasible roots, whereas open squares indicate infeasible roots. (b) Eigenvalues of the Jacobian at the feasible coexistence equilibria, showing local instability for Set A and local asymptotic stability for Set B
Bifurcation governed by the direct transmission threshold $ \mathcal{R}_I $. Panel (a) shows the bifurcation diagram with respect to $ \beta_L $, where $ \beta_L^* = d_I/K $ corresponds to $ \mathcal{R}_I = 1 $. Panels (b) and (c) show the pest-free phase plane dynamics for $ \mathcal{R}_I < 1 $ and $ \mathcal{R}_I > 1 $, respectively
Codimension-two threshold interaction and local dynamics near the candidate Bogdanov–Takens point. (a) The $ (\beta_L, d_V) $-plane divided by the threshold curves $ \mathcal{R}_I = 1 $ and $ \mathcal{R}_V^{(1)} = 1 $, whose intersection gives the codimension-two threshold point. (b)–(e) Local phase portraits near $ E_1 = (K, 0, 0) $ for parameter values below and above the two critical thresholds
Two-parameter region diagram in the $ (\beta_L, d_V) $-plane showing different dynamic regimes of System (2.1)
Time series and phase portraits before, near, and after the Hopf bifurcation point $ d_V^H $
Hopf bifurcation diagram with respect to the pest mortality rate $ d_V $. The critical value $ d_V^H $ separates oscillatory coexistence from stable coexistence
Effects of vector-mediated transmission on phase dynamics and Hopf bifurcation. (a)–(b) show the $ (V, I) $ phase portraits for $ \beta_T = 0.0001 $ and $ \beta_T = 0.0200 $, respectively. (c)–(d) present the Hopf bifurcation diagrams for infected plant biomass $ I $ and pest density $ V $ with respect to $ \beta_T $, with a critical Hopf point at $ \beta_T^H\approx0.005897 $
Effects of allomone-mediated pest suppression under $ \eta_S > \eta_I $. (a)–(b) The responses of $ V(t) $ and $ I(t) $ under weak, moderate, and strong allomone effects; (c) the corresponding stability regions in the $ (\eta_S, \eta_I) $-plane
Two-parameter dynamic regions of Model (2.1). Panels (a)–(f) illustrate the combined effects of the allomone parameters $ \eta_S $ and $ \eta_I $ with $ \beta_T $, $ d_V $, and $ \beta_L $. The colored regions indicate the numerically identified dynamic regimes, while the black curves indicate the corresponding threshold and numerically determined Hopf boundaries
State variables and optimal control profile for the treatment-only strategy: (a) Susceptible plant biomass, (b) infected plant biomass, (c) pest density, and (d) optimal treatment control
State variables and optimal control profile for the insecticide-only strategy: (a) Susceptible plant biomass, (b) infected plant biomass, (c) pest density, and (d) optimal insecticide control
State variables and optimal control profiles for the combined control strategy: (a) Susceptible plant biomass, (b) infected plant biomass, (c) pest density, and (d) optimal treatment and insecticide controls
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