什么是生物数学ppt课件

1、HOPF BIFURCATION IN A SYNAPTICALLY COUPLED FHN NEURON MODEL WITH TWO DELAYS Liping Zhang College of Science, Nanjing University of Aeronautics and Astronautics 2021/07/30.什么是生物数学？生物数学是生物学与数学之间的边缘学科，用数学方法研讨和处理生物学问题，也对与生物学有关的数学方法进展实际研讨。对于今天的生物学者，数学的价值更应该表达在建立在数量化根底上的模型化。经过数学模型的构建，可以将看上去杂乱无章的实验数据整理成有序可

2、循的数学问题，将问题的本质笼统出来。.两个最近的事例SARS在对SARS的研讨中，生物数学就发扬了作用。2003年春SARS迸发时，在有效的疫苗和抗病毒药物研制出来之前，科学家最关怀的是SARS流行的特征。两个国际协作的研讨小组运用了SEIR数学模型，对SARS的传播趋势进展分析和预测，给有关部门提供了参考意见。.Avian Influenza Bird fluAvian influenza is a disease of birds caused by influenza viruses closely related to human influenza viruses.Transmiss

3、ion to humans in close contact with poultry or other birds occurs rarely and only with some strains of avian influenza. The potential for transformation of avian influenza into a form that both causes severe disease in humans and spreads easily from person to person is a great concern for world heal

4、th.Avian Influenza Bird flu.生物数学几个领域的根本引见种群动力学: 种群的相互作用 生物资源管理和综合害虫控制流行病动力学药物动力学生物数学中的斑图生物信息学.生物数学已有一百年多年的历史：1798年Malthus人口增长模型1908年遗传学的Hardy-Weinbe“平衡原理1925年Volterra捕食与被捕食模型1927年KM传染病模型1973年许多著名的生物学杂志相继创刊现如今“生物信息学的诞生是生物数学开展的里程碑.时滞时滞对生物种群的影响不断是生物学家关怀的问题，时滞经常出如今生物的活动中。例如我们日常生活中遇到的视觉和听觉的时滞景象、动物血液再生原理，

5、森林再生原理等。思索到种群密度变化对于增长率的影响都不是瞬间发生的,而是与过去的生活形状有关, 即有时间滞后的，还有动物消化食物也需求一定的时间。在生物数学模型中假设引入时滞，相应的动力系统就变成了带时滞的非线性动力系统。 .由于时滞生物动力系统的演化不仅依赖于系统的当前形状，还依赖于系统过去某一时辰或假设干时辰的形状，其运动方程要用泛函微分方程来描画，和常微分方程系统所描画的系统不同，时滞对系统的动态性质有很大的影响，时滞动力系统普通有无穷多个特征值，解空间是无限维的，其实际分析往往很困难。 .目前，对于非线性时滞动力系统尚没有针对性特别强的研讨方法，讨论非线性常微分方程的方法，大多可以 经

6、过改造用于非线性时滞微分方程的研讨。例如研讨生物动力系统平衡点存在独一性方法有：不动点定理、M-矩阵和重合度实际等；平衡点部分稳定性分析最根本的方法仍是调查特征方程根的变化，例如无害时滞不改动系统正平衡位置的渐近稳定性，所以利用时滞为零时系统的渐近性去研讨时滞不为零时系统正平衡位置的部分稳定性，即用线性近似法研讨研讨平衡点的部分稳定性问题。对小时滞模型用平均法，对常数时滞以及延续时滞模型的全局稳定性主要用Lyapunov方法。研讨分岔景象的常见方法有：中心流形法、规范形实际、Lyapunov-Schmidt方法、摄动法和多尺度法等。 .1.assumptionsThe basic model

7、makes the following assumptions:(H) The model is given by the following system:.Fan D, Hong L. Hopf bifurcation analysis in a synaptically coupled FHN neuron model with delays. Commun Nonlinear Sci Numer Simulat (2021), doi:10.1016/jsns.2021.07.025.2.Stability for FHN neuron model with one delayObvi

8、ously, E (0,0,0,0) is an equilibrium of system (1), linearizing it gives .The characteristic equation associated with system (2) is given byWhere, .For and , Eq.(3) becomes (4)By Routh-Hurwitz criterion we know that if (H) is satisfied then all roots of Eq.(3) have negative real parts. .3.Bifurcatio

9、n for FHN neuron model with one delayObviously, iv(v0) is a root of Eq.(4) if and only if (5)Separating the real and imaginary parts gives (6).Taking square on the both sides of the equations of (6) and summing them up ,and let y=v2 , which leads to:where(7).DenoteThen we have(8)(9)Set (10).Let , th

10、en (10) becomes (11)where , Define (12).Without loss of generality, we assume that Eq.(7) has four positive roots, denoted by , and , respectively. Then Eq.(6) has the four positive roots we have (13).Denote .Where k=1,2,3,4;Then is a pair of purely imaginary roots of Eq.(4) with Similar to the prov

11、es of 8 we know that Eq.(7) has more than one positive roots. Then the stability switch may exist. Summarizing the above discussions we can ensure the stability interval. (14).Theorem 3.1 Suppose that (H) is satisfied and If the conditions (a) (b) , , and (c) , , and there exists a such that and are

12、 not satisfied, then the zero solutions of system ( 1) is asymptotically stable for all .If one of the conditions (a),(b) and (c) of (1) is satisfied, then the zero solution of system (1) is asymptotically stable when If one of the conditions (a),(b) and (c) of (1) is satisfied,and , then the system

13、 (1) undergos a Hopf bifurcation at (0,0,0,0) when .4.Stability and Hopf bifurcation for FHN neuron model with two delayNow let be a root of Eq.(2) Then we get (16)Where .Taking square on the both sides of the equations of (14), we get (15)(15).If Eq.(15) has positive root, without loss of generalit

14、y , we assume Eq.(15) has N positive roots, denoted by 。Notice Eq.12we get(16) .Define .Let be the root of Eq.(4)Satisfying . By computation, we get .WhereSummarizing the discussions above, we have the following conclusions.Theorem 4.1 Suppose that (H), hold and Eq.(14) has positive roots. and have

15、the same meaning as last definition. We get (1) All root of Eq.(4) have negative real parts for and the equilibrium of system (2) is asymptotically stable for .(2) If hold , then system (2) undergos a Hopf bifurcation at the equilibrium E, when . .5.Stability and direction of the Hopf bifurcation In

16、 the previous section, we obtained conditions for Hopf bifurcation to occur when . In this section we study the direction of the Hopf bifurcation and the stability of the bifurcation periodic solutions when , using techniques from normal form and center manifold theory.We assume Letting and dropping

17、 the bars for simplification 17 .Where and (18) Where .From the discussion in Section 2, we know that system (10) undergos a Hopf bifurcation at (0,0,0) when , and the associated characteristic equation of system (10) with has a pair of simple imaginary roots .ResultBased on the above analysis, we c

18、an see that each gij can be determined by the parameters. Thus we compute the following quantities:. (29) .Theorem 5.1. In (29), determines the direction of Hopf bifurcation; if , then the Hopf bifurcation is supercritical (subcritical) and the bifurcation periodic solution exist for ; determines the stability of the bifurcation period solution; bifurcating periodic solution are stable