An "adaptive" approach to control explosive aphid populations

TL;DR

This work tackles the complex, multi-peak dynamics observed in aphid populations by examining three modeling approaches. It first analyzes a variable carrying capacity logistic model (VCM) that can reproduce multiple peaks but may exhibit finite-time blow-up, then introduces an adaptive-behavior switch to bound growth while preserving multi-peak patterns, and finally couples the dynamics to abiotic drivers via a non-autonomous, time-varying fitness framework. The authors provide rigorous results and numerical illustrations showing blow-up in the VCM, the stabilizing effect of the adaptive switch, transient yet non-sustained periodicity under environmental forcing, and a consistent ET/EIL-based assessment indicating the need for management across all models. Collectively, the work offers a stable, biologically grounded prediction scheme for pest outbreaks and informs practical control strategies within an integrated pest management context.

Abstract

Classical models of aphid population dynamics are unable to explain multi-peak patterns in field populations. We consider the variable carrying capacity model (VCM), which can generate such complex multi-peak dynamics, but is also demonstrated to show finite-time blow-up behavior via a sign switching structural instability. We build an adaptive behavioral model with a density-dependent switch to stabilize growth, effectively eliminating blow-up, and also capable of generating multiple peaks. Furthermore, guided by empirical work on environment drivers for pests, we devise a non-autonomous model with time-dependent host plant fitness, successfully connecting transient population dynamics with abiotic drivers such as flooding. Finally, we discuss the practical significance of the results through the Economic Threshold (ET) and Economic Injury Level (EIL) calculation for all models. Our simulations all clearly show that aphid abundances exceed these threshold levels, and control is required. Our work provides a stable and biologically relevant prediction scheme for pest outbreaks and their management strategy.

Figures (8)

• Figure 1: For Figure \ref{['fig:kcm_boombust']}: The time series figure shows the boom-bust scenario for model \ref{['eq:cl1']}. The parameter set used: $a = 0.000005, r=0.3$. The initial population density is $h_0=0,x_0=10$. For Figure \ref{['fig:klm_x0_80']}: The time series figure shows the carrying capacity and aphid population density with $t = 80$ for the Logistic population model with variable carrying capacity \ref{['eq:1']}. The parameter set used: $K_{max} = 10000, K_{min} =1, d =.033, a = 0.000005, r=0.3$. The initial aphid population density is $x_0 = 10$. For Figure \ref{['fig:klm_x0_85']}: The time series figure shows the carrying capacity and finite time blow-up of the aphid population density at $t \approx 85$ for the Logistic population model with variable carrying capacity \ref{['eq:1']}. The parameter set used: $K_{max} = 10000, K_{min} =1, d =.033, a = 0.000005, r=0.3$. The initial aphid population density is $x_0 = 10$.
• Figure 2: The time series figure shows the carrying capacity and finite time blow-up in the aphid population with variation in the initial population. The parameter set: $K_{max} = 10000, K_{min} =1, d =.033, a = 0.000005, r=0.3$. The aphid population blows up at: Figure \ref{['fig:sfig1']}$x_0 = 40$, $t \approx 85$ with $x \approx 2000$, Figure \ref{['fig:sfig2']}$x_0 = 150$, $t \approx 28$ with $x \approx 4800$, Figure \ref{['fig:sfig3']}$x_0 = 500$, $t \approx 23$ with $x \approx 6800$.
• Figure 3: The time series figure shows the carrying capacity and finite time blow-up with variation in $"a"$. The parameter set: $K_{max} = 10000, K_{min} =1, d =.033, r=0.3$. For Figure \ref{['x0_105']}$a = 0.000005$, $x_0 = 105$ and population blows up at $t \approx 30$, Figure \ref{['x0_10']}$a = 0.00005$, $x_0 = 10$ and population blows up at $t \approx 27$, Figure \ref{['x0_1']}$a= 0.0005$, $x_0 = 1$ and population blows up at $t \approx 30$.
• Figure 4: Multiple peaks in the aphid population can be seen for the combined model \ref{['model:part1']}-\ref{['model:part2']}. The combined model was simulated for a time span of 90 days, where $t_{ub}=85.2$ is when the switch was made. Until this time step $t_{ub}$, the model \ref{['model:part1']} was simulated, and then from the next time step until $t_{end}=90$, the model represented by equations \ref{['model:part2']} was simulated. The initial populations at $t=0$ were chosen as $h(0)=0,x(0)=10,$ and then the initial condition for model \ref{['model:part2']} was taken from the populations at the last evaluated time step so, $h_{ub}=0.5040, x_{ub}=7632.6999$. The parameters used are $K_{max} =10000, K_{min} =1, d =.033, a = 0.000005, r=0.3.$
• Figure 5: For Figure \ref{['f1']}: The time series of the cumulative aphid population $h(t)$, the soybean aphid population $x(t)$ and the time dependent parameters $a(t)$ and $r(t)$ are shown in the figure for model \ref{['am1']}. The parameters and initial conditions taken fixed for all the figures are $r_0=0.3, a_0=0.005, \ \omega=q\pi,\ q=0.3,\ x(0)=0.5, \ h(0)=0, r(t)= r_0 (1+\sin{\omega t}), a(t)= a_0 (1+\sin{\omega t})$. It is seen that the aphid population shows a transient periodic solution with multiple peaks, with both $r(t)$ and $a(t)$ being time-dependent. For Figure \ref{['f2']}: Time series of the cumulative function $h(t)$ (orange curve) along with the horizontal threshold line $\sup r(t) = 0.6$ (red dashed). The vertical dashed line indicates the critical time $T^* \approx 28.09$, at which $h(t)$ first exceeds $\sup r(t)$. This marks the end of the transient periodic regime and the beginning of monotonic decay in the population $x(t)$. For Figures \ref{['f3']}--\ref{['f4']}: Simulations when both parameters are taken as constants, i.e., $r(t)=r_0$ and $a(t)=a_0$. In this case, the system does not exhibit any transient periodic dynamics.

Theorems & Definitions (41)

• Remark 1
• Remark 2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• proof