MATH3104: Populations@Play Anthony J. Richardson.
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MATH3104: Populations@Play Anthony J. Richardson
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Transcript of MATH3104: Populations@Play Anthony J. Richardson.
MATH3104: Populations@Play
Anthony J. Richardson
Anthony J. Richardson
Going too slow through this. 2 lectures up to p. 10. Also, need to spend longer preparing for each lecture - 2 hrs is needed. In 2nd lecture on this stuff did heaps of work on board - need to be a bit more confident of working out the maths without notes - practice! Seems pretty easy finding turnign point of logistic natural production function. Need to be able to convey to students where we are going better...Wildebeest Prac - went pretty well - gave them no code this year - generally did well but ran out of time - need to make sure no one after us in prac next year, as we are being kicked out exactly at 4 pm. need to finish 15 mins early to wind up with class. Should have showed them how to do things in Matlab on the screen at front (take laptop) and show examples of things but not the actual programme. e.g. plotting population sizes x harvest levels vs years with plot and Legend command to get different line types and colours. e.g. derivign random numbers. Also, in 1st lab need to show them things like - Matlab working directory, saving the file, how to run, why I highlighted in class, etc...
Anthony J. Richardson
Would be good to finish off with a summary slide of the different models we covered and put them in some biological perspective. e.g. we built up models from simple population models to more realistic predator-prey relationships. Would be good if we used symbolic toolbox - student said can get Mathematica symbolic analysis on web for free - could use that. Also, populations try and avoid chaos region. How do you know which model to use in reality - often start simple and get more complex. Would have been good if we covered interspecific competition - would need to leave something out to cover this. Maybe leave out fitting data to models (Gauss experiments), although we do that in the prac (maybe good reason to leave out) - it might also come back with interspecific competition.Prac 3. nonlin fit doesn't work in lab - requires stats toolbox which I have but not in lab. Lab has lsqcurvefit, which does the same thing.lsqcurvefit(Modelfun,initial,n,delta_n)in our example. Replace in notes. Also pracs require an explanation from me a bit at start to contextualise what we are doing (make learning more efficient). Also, need me to plug in my laptop and show them things on screen altogether, especially at start when I need to show things like saving functions. I also need to give them a handout or website showing matlab functions such as if then, for loops etc. Should put tutes and lectures up on web at start of my teaching so they can print out in the size they want. Q4 in Tute 3 should also go for 500 or 1000 iterations to see the U-shape for chaos properly. Would be good to put a conclusion in about most populations avoiding chaos for tute 3.Tute 4: Went pretty well. Should have shown them debugger on a big screen so didn't have to do it row by row. Need to fix the "bug" in the tute - Roughgarden has coded an geometric model rather than a logistic which would need a Nt+delta_n on the RHS whereas he only has the delta_n I think. Note that K=8000, and the population only gets to 4000, even when the parasitoid is absent at the start (before intro). I could not get it to work properly... Tricky bit is finding interesting parameter values. Need to explain the aim a bit better - keep some parasitoid around - don't want both going extinct, because hosts will come back and no parasitoids around so hosts go crazy. Thus want parsitoid at low numbers. Do I want extinct in code? Just needs a bit more thought. Duncan thought it was good to get them to comment the code - do that for each tutorial???Probably need to understand the theory behind and implications of this last lecture a bit better also...
Outline
• Investigate predator-prey interaction– without density dependence of prey– with density dependence of prey– with predator satiation and density
dependence of prey
• Focus on cycles and equilibria
Predator-Prey Interaction• Many big predators but most small
• Predators often liked more than prey???
• Questions– How many predators can prey sustain?– How much predation can the prey tolerate?– Are there cycles?
Volterra Predator-Prey Model
• 1st predator-prey model in ecology by Vito Volterra in 1920s
• Prey grow exponentially in absence of predator; predators die exponentially in absence of prey
• No density dependence
• Predators contact prey by “random collision”
Volterra Predator-Prey Model
2111 NaNrN
dt
dN
2212 dNNbaN
dt
dN
1N
2Nr
ab
Number of prey
Number of predators
Intrinsic rate of prey increase
Coefficient relating prey capture to predator-prey collisions
Coefficient relating prey capture to predator births
d Death rate of predator
Volterra Predator-Prey Model
• What does this model tell us?
• What are equilibria of these equations?
• What does your answer mean?
• Let’s plot system of equations and see what it looks like
0 500 1000 1500 2000 2500 3000 35000
20
40
60
80
100
120
140
Prey abundance
Pre
da
tor
ab
un
da
nc
e
nhat
nhat/4nhat/2
Volterra Predator-Prey Model
• 4 main lessons1. Predators can control exponential prey
growth
2. Any prey population can support a predator population
3. Formula for equilibrium abundances give estimate of average abundances through time
4. Predator-prey interaction has tendency to oscillate
Adding Density Dependence to Prey
• How big does K of prey have to be to support the predator?
211
11 )1( NaN
K
NrN
dt
dN
2212 dNNbaN
dt
dN
ba
dN 1ˆ
)1(ˆ2 baK
d
a
rN
Volterra Predator-Prey with Density Dependence
• Let’s plot system and see what it looks like• Density dependence has large effect on dynamics• Value of K critical for coexistence
0 200 400 600 800 1000 1200 1400 1600 1800 20000
10
20
30
40
50
60
70
80
Prey abundance
Pre
da
tor
ab
un
da
nc
e
ba
dK ba
dK
ba
dK
0 200 400 600 800 1000 1200 1400 1600 1800 2000-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Prey abundance
Pre
da
tor
ab
un
da
nc
e
0 200 400 600 800 1000 1200 1400 1600 1800 20000
20
40
60
80
100
120
140
160
180
Prey abundance
Pre
da
tor
ab
un
da
nc
e
No Predators
Stable, but no cycles
Stable focus, spiral cycles
Volterra Model with Predator Satiation and Density Dependence
• Term for feeding is
• Can use non-linear feeding term – e.g.
2121 )( NaNNaN
Linear feeding term
)1(1
c
aN
ec
What does this feeding term look like?
Volterra Model with Predator Satiation and Density Dependence
21
11 )1()1(
1
NecK
NrN
dt
dN c
aN
222 )1(
1
dNNebcdt
dN c
aN
0 500 1000 1500 2000 2500 3000 3500 40000
50
100
150
200
250
Prey abundance
Pre
da
tor
ab
un
da
nc
e
0 500 1000 1500 2000 2500 30000
20
40
60
80
100
120
140
160
180
200
Prey abundance
Pre
da
tor
ab
un
da
nc
e
0 500 1000 1500 2000 2500 30000
10
20
30
40
50
60
70
80
90
100
Prey abundance
Pre
da
tor
ab
un
da
nc
e
K = 1000, Stable node K = 2000, Stable focus K = 4000, unstable focus surrounded by limit cycle
What does this system of equations look like?
