Population Dynamics Part III

1
Population Dynamics Part III

• Delay models

2
Deficiency of all models shown thus far

• Birth rate was considered to act instantly, based
  on the present population size.
• For many species there is a considerable time
  delay (pregnancy, juveniles reaching maturity
  etc.)
• A more reasonable model should assume that
  present birth rate (at time t) actually depends
  on y(t-Td) and not on y(t), where Td is a known
  pure time delay.

3
Delay may cause an oscillatory behavior
Example (simple linear equation)
Above can be verified by direct substitution
4
Linear Delay Model
We shall not dwell on the mathematics Is the
following true?
Well study this case later on in simulations only
5
Logistic Delay Model
In this model the regulatory effect depends on
the population at an earlier time
6
Logistic Delay Model Normalized
7
Logistic Delay Model Simulation Results I
If DrTd lt p/2 there is no limit cycle (i.e. a
steady oscillation). Population y goes to K.
8
Logistic Delay Model Simulation Results II
If DrTd gt p/2 there is a limit cycle about yK
with a period T which is in the order of
magnitude of T4Td
9
Logistic Delay Model Simulation Results III
Example 1 DrTd 1.6 then Oscillation period
T4.03Td and ymax/ymin2.56
10
Logistic Delay Model Simulation Results IV
Example 2 DrTd 2.1 then Oscillation period
T4.54Td and ymax/ymin42.3
11
Logistic Delay Model Simulation Results V
Example 3 DrTd 2.5 then Oscillation period
T5.36Td and ymax/ymin2930
12
Comparison of Nicholsons (1957) experimental
data for the population of the Australian
sheep-blowfly and a logistic delay model with
DrTd2.1
From Murrays book

• Observed oscillation period was 35-40 days.
• Model was fit using appropriate K,r,Td values.

13
Comparison of Nicholsons (1957) experimental
data for the population of the Australian
sheep-blowfly and a logistic delay model with
DrTd2.1

• Model implied Td9 days (using the relationship
  T4.54Td. Actual time for larva to mature into
  adult is around 11 days.
• Oscillation is actually independent of K (the
  amount of food available to the insects)

14
Can a First-order scalar equation without delay
have a periodic solution?
Here is a general first-order equation
Lets see if a periodic solution y(t)p(t)p(tT)
works
15
First-order scalar equation without delay cannot
possibly have a periodic solution
Proof
But the left-hand side is positive. Therefore
periodic solution is impossible. The delay plays
a crucial role in the periodicity of the solution.