A Nonlinear StageStructured Cannibalism Model
A Nonlinear StageStructured Cannibalism Model
A cannibalism model published in [1] is described by a system of difference equations:
x
1,t+1
x
2,t

β
1
x
2,t
e
μ
1
x
1,t

β
2
x
2,t
e
x
2,t+1
x
1,t

β
3
x
2,t
e
μ
2
x
2,t
where and are the immature and mature parts of the population at time , respectively, (density independent) is the fecundity (number of newborns per adult), and is the fraction (density independent) of the immature population that survives and enters the mature stage one unit time later (). The parameters and may be interpreted as the natural death rates (not the fishing mortality) caused by factors other than cannibalism. The nonlinear interactions in the model are of the Ricker type and the corresponding parameters ≥0,i=1,2,3 will be referred to as the cannibalism parameters. The model has a large parameter region with stable equilibrium points. It may transition from stable to unstable through a (supercritical) Hopf bifurcation, which means that beyond the instability threshold the dynamics are a quasiperiodic orbit restricted to an invariant curve that surrounds the unstable equilibrium and ).
x
1,t
x
2,t
t
F
P
0<P≤1
μ
1
μ
2
β
i
(
*
x
1
*
x
2