Figure 4

The function \(s_{i}(x)\) has the Ricker-type form such that \(s_{i}(x)=a_{i}e^{-x/k_{i}}\), \(i=1,2,4\), with the parameters \(a_{1}=0.7\), \(a_{2}=0.4\), \(s_{3}=0.68\), \(a_{4}=0.2\), \(k_{1}=300\), \(k _{2}=350\), \(k_{4}=500\), \(r_{1}=0.8\), \(r_{2}=0.8\), \(r_{3}=0.8\), As \(b=12\), \({\Re }_{0}=2.1391\). The positive fixed point is globally asymptotically stable, as shown in the upper left figure. As \(b=70\), \({\Re }_{0}=12.4780\). The solution appears to oscillate, as shown in the upper right figure. As \(b=100\), \({\Re }_{0}=17.8257\), and a cycle with period exists, as shown in the lower left figure. As \(b=200\), \({\Re }_{0}=35.6514\), and the system exhibits chaotic behavior, as shown in the lower right figure. All populations shown in the figures are only eggs for clearer views