Research On Biological Optimization Control Problem Based On Impulse Effect

The control with threshold has been widely used to various domains,such as eco-logical system,treatment of disease,gene engineering and pest control,especially the management of aquatic ecosystem and the treatment of cancer diseases.Based on the above issues,we describes biological significance by impulsive differential equations,and gives a reasonable management strategy.The control with threshold has been used widely in the predator-prey model,in which density dependence or cannibalism of predator is absence.Meanwhile,the most of works have discussed a single case that the periodic solution is induced by a global stable interior equilibrium.Furthermore,the periodic solution and bifurcation of impulsive system with the small parameter pertur-bation is rare.In the second section,we investigated a herbivore-plankton interaction model with cannibalism.For this model,firstly,the complex dynamic behaviors in-cluding the number,the existence and stability of internal equilibriums,the limit and homoclinic cycles,Hopf,saddle-node,Pitchfork and Bogdanov-Takens bifurcations are investigated roughly.Furthermore,multiple state feedback controls are implemented so that three order-one periodic solutions and heteroclinic bifurcation driven by disparate dynamics are induced,respectively.By the method of Bendixson theorem and suc-cessor function,the existences of order-one periodic solutions are proved.Finally,for the corresponding small parameter perturbed system,B-convergent of order-one pe-riodic solution and homoclinic bifurcation are investigated by the variational equation and the rotated vector,respectively.Singular intermittent androgen suppression(IAS)in the prostate cancer lacks the impulsive chemotherapy,neglects the residual effect of medicine,the regular times of therapy,which is often relapsed by the increasing of prostate specific antigen(PSA)level during the on-treatment.So,IAS with chemother-apy in pulsed pattern has become an indispensable clinical scheme for prostate cancer.In the third section,an IAS model with chemotherapy is presented to describe the trans-formation mechanism for three kinds of cancer cells.The model is then extended to include the residual effect of chemotherapy(REC)which suppresses the cancer cells production,thereby preventing the relapse.The optimal controls represent the efficien-cy of both IAS and chemotherapy in suppressing relapse of prostate cancer.Based on optimal algorithms,numerical simulations are implemented not only to show the opti-mal durations of on-and off-treatment and chemotherapy dosages but also to present the effectiveness of different strategies in inhibiting the relapse for three patients.Finally,comparing the PSA under IAS schedule with REC to one without REC demonstrates the validity of both our model and algorithms in lessening the PSA and decreasing the chemotherapy dosages.