Forest Defoliators

Mathematical Analysis of an Epidemic‐Species Hybrid Dynamical System

“…Equilibria of model (7) are obtained by solving / = / = / = V/ = 0. It can be checked that model (7) has the following three boundary equilibria: the vanishing equilibrium 0 (0, 0, 0, 0); the disease-free equilibrium 1 (1, 0, 0, 0); and the predator-free equilibrium 2 ( , , 0, V), where = / , = ( − )/ ( + ), and V = ( − )/( + ); when > , the equilibrium 2 is nonnegative. The positive equilibrium * ( * , * , * , V * ) is obtained by solving the following equations:…”

“…So positivity of the real part of the above expression in parenthesis really indicates the orbital stability of the periodic solution arising out of Hopf bifurcation. For model (7) we considered, let the right four equations…”

“…In this section, we use numerical simulations to illustrate the dynamical behaviors of system (7) and (33). We choose as bifurcation parameter; for system (7), the other parameter values are taken as = 0.1, = 15, = 0.06, = 0.1, = 0.05, = 0.5, and = 0.6; for system (33), the other parameter values are taken as = 20, = 10000, = 0.001, = 0.01, = 10, = 3.1, = 0.005, = 0.5, V = 10, = 1, = 2, and = 1.…”

“…Figure 4 illustrates the stability of the positive equilibrium * . Figure 5 is the bifurcation graph of system (7). The virus replication factor is chosen as the bifurcation parameter.…”

“…One reason for this optimism is that the generation time lapse of microbes is much shorter than that of an insect pest [1]. The role of microbial pesticides in the integrated management of insect pests has been recently reviewed for agriculture [2][3][4][5], forestry [6,7], and public health [8]. In most cases no single microbial control agent will provide sustainable control of an insect pest or complex of pests.…”