Chapter 4
Steady State, Oscillation, and Chaos
in Population Dynamics
And then so slight, so delicate is death
That there’s but the end of a leaf’s fall
A moment of no consequence at all.
(Mark Swann)
4.1 The Emergence of Chaos in Population Models
Let us return to the simple population models of Chap. 2, in the absence of
randomness, and explore the behavior of a very basic deterministic model as a
parameter value gets pushed outside the realm that is typically considered in these
models. Denote the size of the population in time period t as N(t) and the net change
in the population size during that period as ΔN. The exogenous parameter influencing the net flow is R. The net flow ΔN updates the stock N:
ΔN ¼ N t þ DT
ð
ÞÀN t
ð Þ:
ð4:1Þ
The “reproductive rule” in this model is
N t þ DT
ð
Þ¼R Ã N t
ð Þ Ã 1 À N t
ð Þ
ð
Þ ,
ð4:2Þ
and consequently
ΔN ¼ R Ã N t
ð Þ Ã 1 À N t
ð Þ
ð
ÞÀN t
ð Þ:
ð4:3Þ
Compare this equation to Eq. (2.1) of Chap. 2 and describe the differences. Also
see the discussion of discrete versus continuous STELLA flow equation in Chap. 2.
A save-disabled version of STELLA and the computer models of this book are available at
www.iseesystems.com/modelingdynamicbiologicalsystems.
B. Hannon and M. Ruth, Modeling Dynamic Biological Systems,
Modeling Dynamic Systems, DOI 10.1007/978-3-319-05615-9_4,
© Springer International Publishing Switzerland 2014
47
Steady State, Oscillation, and Chaos
in Population Dynamics
And then so slight, so delicate is death
That there’s but the end of a leaf’s fall
A moment of no consequence at all.
(Mark Swann)
4.1 The Emergence of Chaos in Population Models
Let us return to the simple population models of Chap. 2, in the absence of
randomness, and explore the behavior of a very basic deterministic model as a
parameter value gets pushed outside the realm that is typically considered in these
models. Denote the size of the population in time period t as N(t) and the net change
in the population size during that period as ΔN. The exogenous parameter influencing the net flow is R. The net flow ΔN updates the stock N:
ΔN ¼ N t þ DT
ð
ÞÀN t
ð Þ:
ð4:1Þ
The “reproductive rule” in this model is
N t þ DT
ð
Þ¼R Ã N t
ð Þ Ã 1 À N t
ð Þ
ð
Þ ,
ð4:2Þ
and consequently
ΔN ¼ R Ã N t
ð Þ Ã 1 À N t
ð Þ
ð
ÞÀN t
ð Þ:
ð4:3Þ
Compare this equation to Eq. (2.1) of Chap. 2 and describe the differences. Also
see the discussion of discrete versus continuous STELLA flow equation in Chap. 2.
A save-disabled version of STELLA and the computer models of this book are available at
www.iseesystems.com/modelingdynamicbiologicalsystems.
B. Hannon and M. Ruth, Modeling Dynamic Biological Systems,
Modeling Dynamic Systems, DOI 10.1007/978-3-319-05615-9_4,
© Springer International Publishing Switzerland 2014
47