| Science and technology progress, the rapid development of human society, makepeople living standards have improved a lot, the demand of humans for the fishalso grow greatly. According to the relevant documentary in2000, the total outputof the artificial fish is the sum total of15years ago. However, with the worseningecological environment, environment pollution is more and more serious, fishliving environment also get great effects, such as, water quality to be contaminatedlead to the nutrient pollution which has more than the ability of the water candegrade. The production and quality to the fish are a serious threat, and cause theextinction of some species of fish. So, researching the fish to absorb nutrients andthe competition relationship between the water and other species, have theextremely vital significance.In establishing the mathematical model of the ecological system described in theprocess, the existence of the delay to species growth rule has more or less influence.Such as, the fish from the nutrient get into their energy needs, the transformationfrom their own needs to produce the next generation and so on, these process whichhave one delay. With the delay nutrients-fish-shellfish mathematical model of thereality of the research process related has more practical meaning, to the actualtrend forecast more accurate.In this paper, the research is the nutrients-fish-shellfish mathematical model withone delay. First, introduced the background of the nutrients-fish-shellfishmathematical model. Second, theoretically proved all the stability of theequilibrium point to the branch parameters, discussed the positive equilibrium ofthe Hopf-bifurcation. Again analyzed the positive equilibrium of the Hopf-bifurcation using the centre manifold and the standard model, get the branchdirection,the branch periodic solutions for stability and the expression of the Hopf- bifurcation. Finally, used the numerical simulation to verify the correctness of theconclusion. |
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