Modeling of Biological Systems

A Mathematical model of Biological Population Growth : Discrete A Mathematical model of Biological Population Growth : A>1 explosion extinction A<1 &

Logistic map : Two fixed points :

General Concepts : Fixed Points : Attractors : One-D. Two-D.

Fixed points in one dimensional space : (MIT OpenCourseWare » Mathematics » Mathematical Exposition, Spring 2005) Fixed points in two dimensional space :

Stable Neutrally Stable Unstable

a=3.5 a=3.8

Bifurcation Diagram : Universality :

Chaos in Mathematical Models Discrete Continuous Logistic map & Population growth & Sleep Oscillator Lorenz & Rossler & Duffing Oscillator &…

A Simple Model of Waking & Sleeping : Discrete A Simple Model of Waking & Sleeping : S(k+1)=.8*R(k)-tansig((1/.01)*R(k))+S(1); R(k+1)=.8*S(k)-tansig((1/.01)*S(k))+R(1); f f -1 S R K S(1) R(1)

S(k+1)=.8*R(k)-tansig((1/.01)*R(k))+S(1); f f f f -1 -1 S R S R P K K S(1) R(1) S(1) R(1) P(n0) S(k+1)=.8*R(k)-tansig((1/.01)*R(k))+S(1); R(k+1)=.8*S(k)-tansig((1/.01)*S(k))+R(1); P(k+1)=.8*R(k)-tansig((1/eps)*S(k))+P(n0); S(k+1)=.8*P(k)-tansig((1/eps)*R(k))+S(1); R(k+1)=.8*S(k)-tansig((1/eps)*P(k))+R(1);

A Model of Convecting Fluids & The Lorenz Equations : Continuous A Model of Convecting Fluids & The Lorenz Equations : 1. the fluid remain stationary fixed points : 2. a circulation pattern that is stable in time 3.the circulating currents & the resulting temperature differences within the fluid start to vary in time

relaxes to the steady non-convective state for r<1 , all of the xyz space is the basin of attraction for the attractor at the origin .

stability analysis : stable node ; r<1 saddle point ; r>1 are stable ; are unstable ;

Simple Signatures of Chaos By Power Spectrum :

Simple Signatures of Chaos Poincare Map & Structure in Phase Space:

Simple Signatures of Chaos Sensitivity to initial conditions & Lyapunov Exponent :

Simple Signatures of Chaos Sensitivity to initial conditions & Lyapunov Exponent :

Simple Signatures of Chaos Lyapunov Exponent :

Simple Signatures of Chaos

Simple Signatures of Chaos

The other Signatures of Chaos : Kolmogorov-Sinai Entropy Fractal Dimension Correlation Dimension Capacity Similarity Dimension Hausdorff Dimension Box counting Dimension

Chaos in the Other Biological Systems dynamics of human illness human cardiac signal human EEG Protein Ion Channels Action Potential ERP signal Pacemaker