Differential Equations and Mathematical Biology
Differential Equations and Mathematical Biology
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DOI：
10.4324/9780203009314
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发表时间：
1983-01
期刊：
影响因子：
--
通讯作者：
D. S. Jones;M. Plank;B. Sleeman
中科院分区：
文献类型：
--
作者：
D. S. Jones;M. Plank;B. Sleeman
介绍人口增长药物管理细胞分裂可分离变量微分方程齐次型方程一阶线性微分方程一阶方程数值解在MATLAB中的符号计算常系数线性常微分方程介绍首先-二阶线性微分方程二阶线性方程求补函数求特定积分振动微分方程n阶线性常微分方程的唯一系统常系数的二阶方程组用一个系统代替一个微分方程一般系统基本系统矩阵表示法初边值问题求解非齐次微分方程线性边值问题数值解模拟生物现象导论心跳神经冲动传递化学反应捕食模型一阶常微分方程系统存在性和唯一性传染病相平面和雅可比矩阵局部稳定性极限环强迫振荡方程组的数值解一阶方程组和高阶方程的符号计算非线性边值问题的数值解附录：存在理论心脏生理学数学局部模型阈值效应相平面分析和心跳模型心跳周期的生理学考虑心脏起搏器模型神经脉冲传输数学兴奋性和重复放电行波行波的定性行为分段线性模型化学反应Belousov-Zhabotinskii反应的波前Fisher方程的相平面分析一般情况下的定性行为螺旋波和λ- omega系统捕食者和猎物捕鱼捕鱼的影响沃尔泰拉-Lotka模型偏微分方程一阶方程的特征二阶线性偏微分方程椭圆型偏微分方程微分方程抛物型偏微分方程双曲型偏微分方程波动方程双曲型方程的典型问题欧拉-Darboux方程解的可视化演化方程热方程变量分离简单演化方程比较定理扩散问题通过膜的扩散能量和能量估计神经脉冲传输的全局行为化学中的全局行为反应图灵扩散驱动的不稳定性和图案形成有限图案形成域分叉和混沌分叉极限环的分叉离散分叉和倍周期混沌极限环的稳定性庞加莱平面平均数值分叉分析不动点和稳定性路径跟踪与分岔分析跟踪稳定极限环离散系统中的分岔奇异吸引子与混沌局部稳定性分析微分方程肿瘤生长引言肿瘤生长的数学模型I基于模型I的球形肿瘤生长的稳定性基于模型I的球形肿瘤生长的数学模型II基于模型II的球形肿瘤生长的稳定性基于模型II的肿瘤生长的流行病Kermack-McKendrick模型疫苗接种孵化模型空间传播选择练习的答案索引
Introduction Population growth Administration of drugs Cell division Differential equations with separable variables Equations of homogeneous type Linear differential equations of the first order Numerical solution of first-order equations Symbolic computation in MATLAB Linear Ordinary Differential Equations with Constant Coefficients Introduction First-order linear differential equations Linear equations of the second order Finding the complementary function Determining a particular integral Forced oscillations Differential equations of order n Uniqueness Systems of Linear Ordinary Differential Equations First-order systems of equations with constant coefficients Replacement of one differential equation by a system The general system The fundamental system Matrix notation Initial and boundary value problems Solving the inhomogeneous differential equation Numerical solution of linear boundary value problems Modelling Biological Phenomena Introduction Heartbeat Nerve impulse transmission Chemical reactions Predator-prey models First-Order Systems of Ordinary Differential Equations Existence and uniqueness Epidemics The phase plane and the Jacobian matrix Local stability Stability Limit cycles Forced oscillations Numerical solution of systems of equations Symbolic computation on first-order systems of equations and higher-order equations Numerical solution of nonlinear boundary value problems Appendix: existence theory Mathematics of Heart Physiology The local model The threshold effect The phase plane analysis and the heartbeat model Physiological considerations of the heartbeat cycle A model of the cardiac pacemaker Mathematics of Nerve Impulse Transmission Excitability and repetitive firing Travelling waves Qualitative behavior of travelling waves Piecewise linear model Chemical Reactions Wavefronts for the Belousov-Zhabotinskii reaction Phase plane analysis of Fisher's equation Qualitative behavior in the general case Spiral waves and lambda - omega systems Predator and Prey Catching fish The effect of fishing The Volterra-Lotka model Partial Differential Equations Characteristics for equations of the first order Another view of characteristics Linear partial differential equations of the second order Elliptic partial differential equations Parabolic partial differential equations Hyperbolic partial differential equations The wave equation Typical problems for the hyperbolic equation The Euler-Darboux equation Visualization of solutions Evolutionary Equations The heat equation Separation of variables Simple evolutionary equations Comparison theorems Problems of Diffusion Diffusion through membranes Energy and energy estimates Global behavior of nerve impulse transmissions Global behavior in chemical reactions Turing diffusion driven instability and pattern formation Finite pattern forming domains Bifurcation and Chaos Bifurcation Bifurcation of a limit cycle Discrete bifurcation and period-doubling Chaos Stability of limit cycles The Poincare plane Averaging Numerical Bifurcation Analysis Fixed points and stability Path-following and bifurcation analysis Following stable limit cycles Bifurcation in discrete systems Strange attractors and chaos Stability analysis of partial differential equations Growth of Tumors Introduction Mathematical model I of tumor growth Spherical tumor growth based on model I Stability of tumor growth based on model I Mathematical model II of tumor growth Spherical tumor growth based on model II Stability of tumor growth based on model II Epidemics The Kermack-McKendrick model Vaccination An incubation model Spreading in space Answers to Selected Exercises Index