In this paper, the authors study the following predator-prey system with weak Allee effect \[ \begin{aligned} \dot x(t)=&rx\left(1-\frac{x}{K}\right)\frac{x}{x+A}-\frac{mxy}{a+x^2},\\ \dot y(t)=&y\left(-d+\frac{ux}{a+x^2}\right), \end{aligned}\tag{1} \] where \(x(t)\) and \(y(t)\) are the densities of the prey and the predator populations at time \(t\), respectively. The parameters \(a,d,m,r,u, A\) and \(K\) are positive constants in which \(r\) is the intrinsic growth rate of the prey, \(K\) represents the carrying capacity of the prey, \(d\) denotes the natural death rate of the predator, the function \(mx/(a+x^2)\) is the nonmonotonic response function, \(q(x)=x/(x+A)\) is the term describing the weak Allee effect, where \(A\) is defined as a “weak Allee effect constant”.
The global analysis shows that system (1) exhibits numerous kinds of bifurcation phenomena including the saddle-node bifurcation, supercritical Hopf bifurcation and homoclinic bifurcation as the values of parameters vary.