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delay.txt
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TSEC Delayed regulation
One mechanism for population cycles might be if regulation is
\emph{delayed} in time
It takes time for individuals to complete their life cycle
Recall that the life cycle is implicit in our simple models
It takes time for the population to damage its resources or build
up natural enemies
----------------------------------------------------------------------
Time-delayed continuous models
How would change a simple continuous-time model into a (relatively)
simple time-delayed model?
Original model: $\bigeq \frac{dN}{dt} = (b(N)-d(N))N$
Be explicit about time: $\bigeq \frac{dN(t)}{dt} = (b(N(t))-d(N(t)))N(t)$
----------------------------------------------------------------------
CONT Time-delayed continuous models
Where should we add delays? Assume we leave the left-hand side alone
(that's what we're trying to model).
ANS \emph{rates} at time $t$ might depend on past conditions
(population at time $t-\tau$)
ANS \emph{population} at time $t$ is just population at time $t$
ANS that is the population that is experiencing births and deaths
ANS $\bigeq \frac{dN(t)}{dt} = (b(N(t-\tau))-d(N(t-\tau)))N(t)$
----------------------------------------------------------------------
Our model
$\bigeq \frac{dN(t)}{dt} = (b(N(t-\tau))-d(N(t-\tau)))N(t)$
For simplicity, we assume that both rates are delayed by the same
amount of time
More realistic models might have different delays
or delay in only one quantity
or \emph{distributed} delays, so that the rate is some kind of
average
----------------------------------------------------------------------
PSLIDE Arrows with time delay
FIG bd_models/birth.bd.Rout-8.pdf
----------------------------------------------------------------------
Model dynamics
POLL If a population is growing, what will happen as it approaches the equilibrium? | What happens as a growing population approaches equilibrium?
ANS It \emph{keeps} growing
ANS It needs to \emph{pass} the equilibrium and look back in time
before it will stop growing
So what happens in the long term?
----------------------------------------------------------------------
Time-delayed dynamics
FIG bd_models/delay_sims.Rout-0.pdf
----------------------------------------------------------------------
Time-delayed dynamics
FIG bd_models/delay_sims.Rout-1.pdf
----------------------------------------------------------------------
Time-delayed dynamics
FIG bd_models/delay_sims.Rout-2.pdf
----------------------------------------------------------------------
Time-delayed population models
Delayed population models show:
{\bf Damped} oscillations (growing smaller and smaller) for
shorter delays
These could be so small that you wouldn't expect to notice
them
{\bf Persistent} oscillations for longer delays
----------------------------------------------------------------------
Time scales
Oscillations will be bigger (and will switch from damped to
persistent) if the time delay in the model is ``long''
Long compared to what?
ANS It must be something else in the model with units of time
ANS It should have something to do with behaviour near the
equilibrium
ANS In fact, we compare the time delay to the characteristic time
of approach to the carrying capacity (calculated by ignoring
the delays)
----------------------------------------------------------------------
Unitless quantities
The behaviour of any particular delay system is determined by one or
more unitless quantities
Our simple model is controlled by the ratio $\tau/t_c$, where $t_c$
is the characteristic time of approach to the carrying capacity in
the absence of delay
In fact, cycles are persistent when $\tau/t_c > \pi/2$!
----------------------------------------------------------------------
Time-delayed regulation
Time-delayed regulation produces simple cycles
Damped when delay is short \ldots
Persistent when delay is long \ldots
\ldots compared to characteristic time of approach to equilibrium
----------------------------------------------------------------------