Fluctuations and Cycles:
The Intriguing Link of Theory and Empirical Studies
Primary Goals:
• Understand link of theory (mathematical models),
and empirical studies/approaches
• Understand basic processes and associated terminology
• Gain appreciation of complexity of population dynamics and
need for multiple scientific approach
• Appreciate application of theory and empirical studies
Primary Sources: Case (2000), Hutchinson (1978), Ricklefs and Miller (1999),
Turchin (2003)
Why Theory?
The mathematical/theoretical treatment (Lotka/Volterra/Pearl) of population
fluctuations began, coincidentally, with the initiation of a rigorous empirical
approach (Elton)
Only recently, do we find a syntheses of empirical/theoretical approaches:
Turchin’s Complex Population Dynamics perhaps providing the best synthesis
Thanks Becky!
Theory is key: tendency for common phenomenon to be overlooked or
misinterpreted in the absence of a well-known body of theory
The primary contribution of theory was in demonstrating that complex dynamics,
such as cycles, could be caused by simple endogenous mechanisms: this provided a
rationale for developing more clever hypotheses than simply trying to identify
exogenous mechanisms, such as weather.
With the debate of Density Dependence/Independence,
Nicholson started the debate that led to analyses of long-term data sets.
From this, emerged the search for analytical methods, discussed by Mary.
Emergence of nonlinear dynamic models, and the, once again,
link between physicists and ecologists, exemplified by Robert May’s work.
Population Dynamics:
Terminology
Trend: long-term (operationally defined) exogenously driven, systematic change
Oscillation: Pop dynamics that have some element of regularity, predictability
Damped Oscillation: oscillations that become less pronounced as they
approach a stable point (e.g., carrying capacity)
Fluctuation: temporal changes in abundance
Irregular fluctuations: irregularity of numbers after trend and endogenous
oscillations have been removed == “env. stochasticity” ==“process noise”
(exogenous env. stochasticity and measurement noise
[sampling and measurement error]
Cycle:
Stable limit cycle: A stable oscillation – never stabilizes at equilibrium
Bifurcation: limit cycle that splits into n-periodicity
Chaos: bounded fluctuations with sensitive dependence on initial conditions
Deterministic Chaos: arises from model without stochasticity
Fluctuations and Cycles:
Two Focuses
Single Species
Population Interactions
Population Oscillations:
Single Species/Spatial Relations
Demography of Single Population
Metapopulation Dynamics
Single-Population Oscillations:
Mechanisms
Endogenous vs Exogenous vs Intrinsic vs Extrinsic Mechanisms
Endogenous: the density-dependent component of population dynamics
Exogenous: the density-independent component; affects density without being
affected by it
Intrinsic factors: pertaining to the focal population; characteristics within
Extrinsic factors: external factors (e.g., predation) affecting focal population
Single-Population Fluctuations:
Mechanisms
First Principle: intrinsic growth rate determines theoretical oscillation potential
Species(populations) with high pop growth rates can track fluctuating env conditions (K)
Single-Population Fluctuations:
Mechanisms
Single-Population Fluctuations:
Mechanisms
Characteristic Return Time, T
Theory suggests a population will track the environment closely when T
is < period of environmental fluctuation/2
AND
When T is >> than period than population varies little
Single-Population Fluctuations:
Mechanisms
Second Principal: Temporal variation in age structure will lead
to oscillatory behavior
Temporal variation often irregular,
leading to fluctuations, not cycles
Single-Population Cycles:
Mechanisms
High R and simple density dependence
Population models developed in 1920-30s demonstrated that populations
subjected to even minor random environmental var could be caused to
oscillate or cycle
Inherent in discrete population models, such as the logistic equation,
N t+1 = (1 + R(1-Nt / K) )
Population Size
N(t)
120
100
80
60
N(t)
40
20
31
28
25
22
19
16
13
10
7
4
1
0
Time
Lets look at what happens to the predicted N with changes in population growth rate, R
Stable Limit Cycles
From Case (2000:115)
R < 2.0
R = 2.0
R = 2.449
R = 2.544
R = 2.564
R = 2.568
Equilibrium point is locally stable
Equilibrium point becomes unstable
a cycle of 2 “is born”
A cycle of period 4 is born
A cycle of period 8 is born
A cycle of period 16 is born
A cycle of period 32 is born
Note: successive period doublings occur faster with inc. R. Converge
on R=2.57, in the limit of an infinite period.
Single-Population Cycles:
Mechanisms
Time delays cause oscillations and cycles in continuous models
Time lags
Continuous time-lag model of Hutchinson (but first used in economics 40 years prior)
dn/dt=rNt ( (K-Nt-T )/K)
The dynamics of this model are controlled by rT; results are sensitive to either r or T
Behavior is well known for relative values of r and T
Chaotic Population Growth
R > 2.57
Chaotic behavior is NOT random. Note the behavior arises
from a deterministic model
Deterministic Chaos: arising from a deterministic model (no stochastic elements),
sensitive to initial conditions, e.g., No.
Note that these behaviors are distinct from the continuous logistic eqn;
the only difference is the lag time of t-interval.
Single-Population Cycles:
Empirical Studies
Single-Population Cycles:
Empirical Studies
Observations and Hypotheses
Three Classes of Hypotheses:
1.
Abiotic (weather and sunspots)
2.
Biotic intrinsic (genotype/phenotypic phys. or behav.)
3.
Biotic extrinsic (food, predation, parasites, disease)
The Chitty Hypothesis:
“all species are capable of limiting their own density without either
destroying the food resources to which they are adapted, or depending
upon enemies or climatic accidents to prevent them from doing so”
“at each mention of the Chitty Hypothesis faculty and students bow
their heads and cross themselves to the accompaniment of religious music”