Exploring bifurcations in a differential-algebraic model of predator–prey interactions

In this work, we first discuss the positive equilibrium point of the continuous predator–prey system and its stability, and we discuss the parameter conditions under which the continuous system undergoes a cusp bifurcation (Bogdanov–Takens bifurcation) of codimension two bifurcation at the positive equilibrium point. Then, we provide an insightful study of discrete predator–prey systems by the use of Euler’s method, which includes square-root function responses and nonlinear prey harvesting. By synthesizing the new standard form of differential-algebraic systems, the central manifold theorem, and the bifurcation theory, we identify the specific conditions under which the system may undergo flip bifurcation and Neimark–Sacker bifurcation. In addition, codimension-two bifurcations associated with 1:2 strong resonances are analyzed by using a series of affine transformations and bifurcation theory. Through numerical simulations, we not only verify the validity and correctness of our findings, but also elucidate the frequency of trajectory bifurcations in the intervals of 2, 4, and 8 and chaotic phenomena. These findings reveal a richer and more diverse dynamic behavior of discrete differential-algebraic bioeconomic systems, which is of great theoretical and practical significance to the fields of mathematics and biology.