Predator Extinction arose from Chaos of the Prey: the Chaotic Behavior of a Homomorphic Two-Dimensional Logistic Map in the Form of Lotka-Volterra Equations
A two-dimensional homomorphic logistic map that preserves features of the Lotka-Volterra equations was proposed. To examine chaos, iteration plots of the population, Lyapunov exponents calculated from Jacobian eigenvalues of the $2$D logistic mapping, and from time series algorithms of Rosenstein and Eckmann et al. were calculated. Bifurcation diagrams may be divided into four categories depending on topological shapes. Our model not only recovered the $1$D logistic map, which exhibits flip bifurcation, for the prey when there is a nonzero initial predator population, but it can also simulate normal competition between two species with equal initial populations. Despite the possibility for two species to go into chaos simultaneously, where the Neimark-Sacker bifurcation was observed, it is also possible that with the same interspecies parameters as normal but with a predator population $10$ times more than that of the prey, the latter becomes chaotic, while the former dramatically reduces to zero with only a few iterations, indicating total annihilation of the predator species. Interpreting humans as predators and natural resources as preys in the ecological system, the above-mentioned conclusion may imply that not only excessive consumption of natural resources, but its chaotic state triggered by an overpopulation of humans may backfire in a manner of total extinction of the human species. Fortunately, there is little chance for the survival of the human race, as isolated fixed points in the bifurcation diagram of the predator reveal. Finally, two possible applications of the phenomenon of chaotic extinction are proposed: one is to inhibit viruses or pests by initiating the chaotic states of the prey on which the viruses or pests rely for existence, and the other is to achieve the superconducting state with the chaotic state of the applied magnetic field.