Dynamic analysis and control of a rice-pest system under transcritical bifurcations

A decision model is developed by adopting two control techniques, combining cultural methods and pesticides in a hybrid approach. To control the adverse effects in the long term and to be able to evaluate the extensive use of pesticides on the environment and nearby ecosystems, the novel decision model assumes the use of pesticides only in an emergency situation. We, therefore, formulate a rice-pest-control model by rigorously modelling a rice-pest system and including the decision model and control techniques. The model is then extended to become an optimal control system with an objective function that minimizes the annual losses of rice by controlling insect pest infestations and simultaneously reduce the adverse impacts of pesticides on the environment and nearby ecosystems. This rice-pest-control model is verified by analysis, obtains the necessary conditions for optimality, and confirms our main results numerically. The rice-pest system is verified by stability analysis at equilibrium points and shows transcritical bifurcations indicative of acceptable thresholds for insect pests to demonstrate the pest control strategy.

where The biologically meaningful equilibria of the system (S16) are the non-negative solutions of 1 ( , ) 0 f x y = and 2 ( , ) 0 f x y = .The rice (prey) isocline consists of the axis 0 x = and the straight line yx  =− and the pests (predator) isocline consists of the axis 0 y = and the line & Petrovskii, 2011).Correspondingly, the system (S16) is bounded by two equilibrium points, the trivial/pre-cultural equilibrium 0 ( , ) (0, 0) E x y = and the pest-free can be found at the intersection of the two isoclines.For the existence of * E , the value of the parameters must follow the conditions 10  + , 0  + and 10  − .
Theorem 2.1.The system (S16) experiences transcritical bifurcation at the pest-free equilibrium point 1 ( , 0) E  as the growth parameter  passes through the critical value *  .Proof At the pest-free equilibrium point 1 ( , 0) E  , the associated Jacobian matrix of the system (S16) takes the form: The set of eigenvalues of 1 * () i.e., one eigenvalue is zero and the other is negative since 0   .Therefore, to examine the nature of the system at 1 E , we have applied Sotomayor's theorem (Perko, 2000).For this purpose, we consider the system (S16) as Let the eigenvectors corresponding to the zero eigenvalues of . Here, Df denotes the partial derivative of f with respect to x and y , and Df  denotes the partial derivative of f with respect to the parameter  .Therefore, * 1 ( ; ) 0 Hence, there is a saddle-node bifurcation at the nonhyperbolic equilibrium point 1 ( , 0) E  at the bifurcation value  .
For 0   , there is no equilibrium point.For 0  = , the 2 1 ( , ) f x y x =− is structurally unstable and the bifurcation value 0  = .Therefore, there is a transcritical bifurcation at the origin for 0  = .There are two equilibria at origin (0, 0) and 1 ( , 0)

Supplemental Information 2.1 -Finding values for dimensionless parameters
We have numerically investigated the dynamic behaviour of the system (S16) for the variation in the growth of pest populations (  ).

Supplemental Information 2.2 -A supportive figure
Figure S3 (A) Phase plane of the rice-pest system (S16) for  = 13.6,(B) time series analysis of (A), (C) phase plane of the system (S16) for  = 13.7, and (D) time series analysis of (C).The system experiences a steady-state limit cycle for  = 13.6, and approaches E1(0,0) for  = 13.7 meaning that the system exticts over a long time.
negative and the other one is zero for the condition 1  =(Perko, 2000).To investigate the nature of the system at * E , we have applied Sotomayor's theorem(Sen, Banerjee   & Morozov, 2012).Let the eigenvectors correspond to the zero eigenvalues of (S16) satisfies all the necessary conditions of Sotomayor's theorem and thus the system (S16) experiences a transcritical bifurcation at the co-existence equilibrium point * E for the bifurcation parameter (Perko,   2000).
Let 0  be the initial condition for the existence of * E .The parameters must follow the conditions 10 (Sen, Banerjee & Morozov, 2012). 2000).