Qualitative Analysis Of The Ecosystem With The Slow-fast Processes

In ecosystem, sometimes the rate of change of populations is different. Some changeslowly, some change rapidly, even they have a big difference. The changes in differentevolution are described very well by the ecosystem model with the slow-fast processes. Basedon the above, crop ecosystem models and forest ecosystem models with slow-fast processesare proposed, respectively. Combined singular perturbation theory with qualitative theory ofdifferential equations, the qualitative analysis is studied. Theoretical foundation is provided toreveal a lot of phenomenon with slow-fast processes in nature. The contents are as follows:Firstly, based on the model with the fold catastrophe, a singular crop ecosystem modeland the corresponding perturbation model are proposed using the principles in biologicaldynamics and ecotoxicology. On the basis of the analysis of the model, we discuss therelation between population ecosystem with slow-fast processes and the singular populationecosystem.Secondly, according to the growth cycle between the tree populations and the insectpopulations, the growth rate of tree populations is considered as slow variable, and thepropagation speed of insect is considered as fast variable, the forest-insect model withslow-fast processes is given.The dynamic behavior of the model is analyzed by singularperturbation method.Thirdly, on the basis of the relation between the energy reserve of the trees and the insectpopulations, the forest-insect model with slow-fast processes is built up. Using the geometricanalysis of singular perturbation method, the slow-fast processes of the model are investigated.The sufficient condition of the existence of limit cycles is given by a separation principle.For the above, the corresponding simulations are carried out to prove the results byMatlab software.