# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021244
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## Crop - Weed interactive dynamics in the presence of herbicides: Mathematical modeling and analysis

 Department of Mathematics, Birla Institute of Technology (BIT), Mesra, Ranchi - 835215, Jharkhand, India

*Corresponding author: Abhinav Tandon

Received  February 2021 Revised  May 2021 Early access October 2021

In the present study, a nonlinear model is formulated to demonstrate crop - weed interactions, when they both grow together on agricultural land and compete with each other for the same resources like sunlight, water, nutrients etc., under the aegis of herbicides. The developed model is mathematically analyzed through qualitative theory of differential equations to demonstrate rich dynamical characteristics of the system, which are important to be known for maximizing crop yield. The qualitative results reveal that the system not only exhibits stability of more than one equilibrium states, but also undergoes saddle - node, transcritical and Hopf bifurcations, however, depending on parametric combinations. The results of saddle - node and transcritical bifurcations help to plan strategies for maximum crop yield by putting check over the parameters responsible for the depletion of crops due to their interaction with weeds and herbicides. Hopf - bifurcation shows bifurcation of limit cycle through Hopf - bifurcation threshold, which supports that crop - weed interactions are not always of regular type, but they can also be periodic.

Citation: Abhinav Tandon. Crop - Weed interactive dynamics in the presence of herbicides: Mathematical modeling and analysis. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021244
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##### References:
Phase diagrams in $(W,C)$ plane with variations in $s$, other parameters same as in $\mathrm{S_1}$
Stability of different equilibria with variations in model parameters of set $\mathrm{S_1}$
Phase - portraits in $(C, W, H)$ - plane for different $\lambda$, other parameters as in $\mathrm{S_2}$
Time - series graphs of system variables for different $\lambda$, other parameters as in $\mathrm{S_2}$
Stability of different equilibria with variations in model parameters of set $\mathrm{S_2}$
Description of Model Parameters
 Parameter Definition $r$ Net growth rate coefficient of crop yield ${K}$ Carrying capacity of crop yield ${\alpha}$ Declination rate coefficient of crop yield due to herbicides ${\beta_1}$ Interspecific rate coefficient of crop yield ${s}$ Net growth rate coefficient of weed yield ${L}$ Carrying capacity of crop yield ${\beta_2}$ Interspecific rate coefficient of weed yield ${\gamma}$ Absorption rate coefficient of herbicides by weeds ${\theta}$ Absorption Efficiency of weeds ${\mu_0}$ Natural alleviation rate of herbicides
 Parameter Definition $r$ Net growth rate coefficient of crop yield ${K}$ Carrying capacity of crop yield ${\alpha}$ Declination rate coefficient of crop yield due to herbicides ${\beta_1}$ Interspecific rate coefficient of crop yield ${s}$ Net growth rate coefficient of weed yield ${L}$ Carrying capacity of crop yield ${\beta_2}$ Interspecific rate coefficient of weed yield ${\gamma}$ Absorption rate coefficient of herbicides by weeds ${\theta}$ Absorption Efficiency of weeds ${\mu_0}$ Natural alleviation rate of herbicides
Coexistence equilibrium and its existence conditions
 Discriminant Values of $\tilde a$ Values of $\tilde b, \tilde c$ Values of $\tilde W$ Feasible Coexistence equilibrium $\Delta_{E_3}<0$ —— —— Complex $\tilde W_1$ & $\tilde W_2$ — $\Delta_{E_3}=0$ $\tilde a>0$ $\tilde b>0,\tilde c>0$ Positive coincident $\tilde W_1$ & $\tilde W_2$ Coincident $E_{3^*}$ exists $\tilde a<0$ $\tilde b<0,\tilde c<0$ $\Delta_{E_3>0}$ $\tilde a>0$ $\tilde b>0,\tilde c>0$ Both $\tilde W_1$ & $\tilde W_2$ are positive Either one or both of $E_{3_1}$ & $E_{3_2}$ may exist $\tilde b=0,\tilde c>0$ Complex $\tilde W_1$ and $\tilde W_2$ — $\tilde b<0,\tilde c>0$ Negative $\tilde W_1$ & $\tilde W_2$ — $\tilde b>0,\tilde c=0$ $\tilde W_1$ is zero & $\tilde W_2$ is positive $E_{3_2}$ exists $\tilde b<0,\tilde c=0$ $\tilde W_1$ is zero & $\tilde W_2$ is negative — $\tilde b>0,\tilde c<0$ $\tilde W_1$ is positive & $\tilde W_2$ is negative $E_{3_1}$ exists $\tilde b=0,\tilde c<0$ $\tilde W_1$ is positive & $\tilde W_2$ is negative — $\tilde b<0,\tilde c<0$ $\tilde W_1$ is positive & $\tilde W_2$ is negative $E_{3_1}$ exists $\tilde a<0$ $\tilde b>0,\tilde c>0$ $\tilde W_1$ is negative & $\tilde W_2$ is positive $E_{3_2}$ exists $\tilde b=0,\tilde c>0$ $\tilde W_1$ is positive & $\tilde W_2$ is negative — $\tilde b<0,\tilde c>0$ $\tilde W_1$ is negative & $\tilde W_2$ is positive $E_{3_2}$ exists $\tilde b>0,\tilde c=0$ $\tilde W_1$ is negative & $\tilde W_2$ is zero — $\tilde b<0,\tilde c=0$ $\tilde W_1$ is positive & $\tilde W_2$ is zero $E_{3_1}$ exists $\tilde b>0,\tilde c<0$ Both $\tilde W_1$ & $\tilde W_2$ are negative — $\tilde b=0,\tilde c<0$ Complex $\tilde W_1$ & $\tilde W_2$ — $\tilde b<0,\tilde c<0$ both $\tilde W_1$ & $\tilde W_2$ are positive Either one or both of $E_{3_1}$ & $E_{3_2}$ may exist
 Discriminant Values of $\tilde a$ Values of $\tilde b, \tilde c$ Values of $\tilde W$ Feasible Coexistence equilibrium $\Delta_{E_3}<0$ —— —— Complex $\tilde W_1$ & $\tilde W_2$ — $\Delta_{E_3}=0$ $\tilde a>0$ $\tilde b>0,\tilde c>0$ Positive coincident $\tilde W_1$ & $\tilde W_2$ Coincident $E_{3^*}$ exists $\tilde a<0$ $\tilde b<0,\tilde c<0$ $\Delta_{E_3>0}$ $\tilde a>0$ $\tilde b>0,\tilde c>0$ Both $\tilde W_1$ & $\tilde W_2$ are positive Either one or both of $E_{3_1}$ & $E_{3_2}$ may exist $\tilde b=0,\tilde c>0$ Complex $\tilde W_1$ and $\tilde W_2$ — $\tilde b<0,\tilde c>0$ Negative $\tilde W_1$ & $\tilde W_2$ — $\tilde b>0,\tilde c=0$ $\tilde W_1$ is zero & $\tilde W_2$ is positive $E_{3_2}$ exists $\tilde b<0,\tilde c=0$ $\tilde W_1$ is zero & $\tilde W_2$ is negative — $\tilde b>0,\tilde c<0$ $\tilde W_1$ is positive & $\tilde W_2$ is negative $E_{3_1}$ exists $\tilde b=0,\tilde c<0$ $\tilde W_1$ is positive & $\tilde W_2$ is negative — $\tilde b<0,\tilde c<0$ $\tilde W_1$ is positive & $\tilde W_2$ is negative $E_{3_1}$ exists $\tilde a<0$ $\tilde b>0,\tilde c>0$ $\tilde W_1$ is negative & $\tilde W_2$ is positive $E_{3_2}$ exists $\tilde b=0,\tilde c>0$ $\tilde W_1$ is positive & $\tilde W_2$ is negative — $\tilde b<0,\tilde c>0$ $\tilde W_1$ is negative & $\tilde W_2$ is positive $E_{3_2}$ exists $\tilde b>0,\tilde c=0$ $\tilde W_1$ is negative & $\tilde W_2$ is zero — $\tilde b<0,\tilde c=0$ $\tilde W_1$ is positive & $\tilde W_2$ is zero $E_{3_1}$ exists $\tilde b>0,\tilde c<0$ Both $\tilde W_1$ & $\tilde W_2$ are negative — $\tilde b=0,\tilde c<0$ Complex $\tilde W_1$ & $\tilde W_2$ — $\tilde b<0,\tilde c<0$ both $\tilde W_1$ & $\tilde W_2$ are positive Either one or both of $E_{3_1}$ & $E_{3_2}$ may exist
Stability behavioral change with values of $s$ for parameter set $\mathrm{S_1}$
 Values of $s$ $E_0$ $E_1$ $E_2$ $E_3$ $0s_{_{T_{E_2}}}$ unstable unstable locally - asymptotically stable non - feasible
 Values of $s$ $E_0$ $E_1$ $E_2$ $E_3$ $0s_{_{T_{E_2}}}$ unstable unstable locally - asymptotically stable non - feasible
Stability behavioral change with values of $\lambda$ for parameter set $\mathrm{S_2}$
 Values of $\lambda$ $E_0$ $E_1$ $E_2$ $E_3$ $0<\lambda<\lambda_{T_{E_2}}$ unstable unstable stable non - feasible $\lambda=\lambda_{T_{E_2}}$ unstable unstable non-hyperbolic (locally asymptotically stable) non -feasible $\lambda_{T_{E_2}}<\lambda< \lambda_{h_1}$ unstable unstable unstable $E_{3_1}$ is locally asymptotically stable $\lambda=\lambda_{h_1}$ unstable unstable unstable first Hopf - bifurcation occurs around $E_{3_1}$, periodic solutions start $\lambda_{h_1}<\lambda<\lambda_{h_2}$ unstable unstable unstable $E_{3_1}$ exhibits unstable periodic oscillations (stable limit cycle) $\lambda=\lambda_{h_2}$ unstable unstable unstable second Hopf - bifurcation occurs around $E_{3_1}$, periodic solutions end $\lambda>\lambda_{h_2}$ unstable unstable unstable $E_{3_1}$ is locally asymptotically stable
 Values of $\lambda$ $E_0$ $E_1$ $E_2$ $E_3$ $0<\lambda<\lambda_{T_{E_2}}$ unstable unstable stable non - feasible $\lambda=\lambda_{T_{E_2}}$ unstable unstable non-hyperbolic (locally asymptotically stable) non -feasible $\lambda_{T_{E_2}}<\lambda< \lambda_{h_1}$ unstable unstable unstable $E_{3_1}$ is locally asymptotically stable $\lambda=\lambda_{h_1}$ unstable unstable unstable first Hopf - bifurcation occurs around $E_{3_1}$, periodic solutions start $\lambda_{h_1}<\lambda<\lambda_{h_2}$ unstable unstable unstable $E_{3_1}$ exhibits unstable periodic oscillations (stable limit cycle) $\lambda=\lambda_{h_2}$ unstable unstable unstable second Hopf - bifurcation occurs around $E_{3_1}$, periodic solutions end $\lambda>\lambda_{h_2}$ unstable unstable unstable $E_{3_1}$ is locally asymptotically stable
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