1D Dynamics
1. MS p 38, Exercise 3. For more practice, find the fixed points of the following models. If there are no fixed points, say so. For each fixed point,
analyse its stability using the deriva
MATH 3MB3 FALL 2016 HOMEWORK 1
due Thursday September 22 at 11:59PM
Format your report using some form of word processing software (Word, Latex, OpenOffice, .),
export it to a PDF file and submit it v
MATH 3MB3 FALL 2016 LECTURE NOTES
linear univariate discrete deterministic models
fixed point(s) of a model N (t + 1) = f (N (t)
by solving the equation N = f (N ). When f
is a linear function, this c
MATH 3MB3 FALL 2016 SAMPLE FINAL 1
Question 1. Consider the following population model:
dx
= rx 1 ebx
dt
where x(t) is the population size, t is time, and r and b are both positive non-zero constants.
MATH 3MB3 FALL 2016 LECTURE NOTES
nonlinear univariate discrete deterministic models
The first equilibrium N = 0 has f 0 (N ) = 1+R
so it is stable if |1 + R| < 1 which is the
same as saying 2 < R < 0
MATH 3MB3 FALL 2016 HOMEWORK 4
due Friday November 11 at 11:59PM
Format your report using some form of word processing software (Word, Latex, OpenOffice, .),
export it to a PDF file and submit it via
MATH 3MB3 FALL 2016 LECTURE NOTES
nonlinear multivariate discrete deterministic models
so that N (t + 1) 6= N (t). If b = 0 and = 0,
so that there are no birth or deaths, the total
population will rem
MATH 3MB3 FALL 2016 LECTURE NOTES
multivariate stochastic models
Explicit solution
One type of multivariate stochastic model is the
Markov chain. Although these discrete models
are reminiscent of cert
MATH 3MB3 FALL 2016 HOMEWORK 2
due Saturday October 8 at 11:59PM
Format your report using some form of word processing software (Word, Latex, OpenOffice, .),
export it to a PDF file and submit it via
MATH 3MB3 FALL 2016 LECTURE NOTES
nonlinear multivariate continuous deterministic models
Lets look at each fixed point. For both M =
N = 0 and M = N = 1, we get
The basic model is
d~x
= f~(~x),
dt
a/2
MATH 3MB3 FALL 2016 LECTURE NOTES
linear multivariate discrete deterministic models
The basic model is ~x(t + 1) = A~x(t). Here Time-dependent solution
~x(t) = (x1 (t), . . . , xn (t) is a vector with
Univariate linear discrete-time deterministic models
Ben Bolker and Steve Walker: September 11, 2017
Basic model: N (t + 1) = f (N (t), where f is a linear
function and t is an integer. Typically, sta
philosophical/armwaving material
Ben Bolker
2017-09-05 11:09:09
Philosophy
Modeling is applied math; mapping between the real world
and mathematical framework. Getting the mapping right
is the hardest
Math 3MB3 Fall 2017
MATH 3MB3 final sample questions
Special Instructions:
Casio FX-991 MS or MS Plus calculator allowed, no other external aids
Stability condition for 2D discrete-time systems: sta
1D Dynamics
1. MS p 38, Exercise 3. For more practice, find the fixed points of the following models. If there are no fixed points, say so. For each fixed point,
analyse its stability using the deriva
Discrete-time Lotka-Volterra model:
Vt+1 = rVt aPt Vt
Pt+1 = sPt + acPt Vt
V is prey (victim), P is predator. r is prey growth rate; a is attack rate
of predators; c is conversion efficiency of predat
MATH 3MB3 FALL 2016 LECTURE NOTES
nonlinear univariate continuous deterministic models
The general formula associated with these mod- Example: constant harvest model
els is
The model is defined by
dx(
MATH 3MB3 FALL 2016 LECTURE NOTES
univariate stochastic models
If P(A|B) = P(A), then A is independent
of B. Independence is equivalent to
In many applications of mathematical modelling, the change i
MATH 3MB3 FALL 2016 LECTURE NOTES
linear univariate continuous deterministic models
case is that overshooting and oscillations do not
occur.
The basic model is
dx(t)
= rx(t),
dt
Explicit solution in t
In class midterm practice
short answer
Time is continuous in reality. Nevertheless, discrete-time models provide good approximations
to continuous time models. Describe three scenarios where this appr
Using R to analyze continuous models (lab 4)
Steve Walker
October 6, 2014
Licensed
under
the
Creative
Commons
attribution-noncommercial
license
(http:/
creativecommons.org/licenses/by-nc/3.0/).
Please
Matrix models in R (lab 3)
c 2010 Ben Bolker (modied by Steve Walker)
September 4, 2014
Licensed
under
the
Creative
Commons
attribution-noncommercial
license
(http:/
creativecommons.org/licenses/by-nc
Consider the following system of ODEs:
dxdt=fi(x1,xn;), i=1n and =(1,m).
Example: x=x2+
If >0: x>0 for x, so that we don't have any equilibria
If <0:
there exists a non-permitted zone in the solution
Literature: Murray, J.D. "Mathematical biology. I. Introduction", Sec. 2.1-2.3
In general, we can define the sequence cfw_xn in the form of a mapping
xn+1=f(xn)
where xn can be some relative quantity
Problem 1 (Taylor formula)
Expand the function f(x)=(cosh(x)sin(x) till o(x5)
Some hints
please take a look The Taylor formula notes and be careful with o(xk),
remember how you dealt with a(x)b(x) kin
Models of a single population growth
Literature: Murray, J.D. "Mathematical biology. I. Introduction", Sec. 1.1
Exponential growth, Logistic equation
N(t) is a size of the population at moment t. We c
Main equation
We assume that the population size is sufficiently large.
dxidt=j=0nxjfjqjixi, i=0n,
where fi is the fitness of the sequence i, qji is mutation probabilities, referring to the production
Stability analysis
Consider the following dynamical system
xi=fi(x1,xn,t), i=1,n
such that the right hand sides are sufficiently smooth in the domains of definition of functions fi regarding their
arg
Plant growth schedule
Literature: Iwasa, Yoh and Dan Cohen (1989) "Optimal Growth Schedule of a Perennial Plant" The American
Naturalist, 133(4), pp. 480-505 (link)
Suppose that we have a plant and it
Stability of dynamical systems
1. Find equilibrium points, give their classification (un)stable focus, (un)stable knot, saddle, center) and plot the
phase trajectories
(A) cfw_x=3x,y=2x+y.
(B) cfw_x=x