Fluctuations and Cycles: The Intriguing Link of Theory and Empirical Studies

Primary Sources: Case (2000), Hutchinson (1978), Ricklefs and Miller (1999), Turchin (2003)

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

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, predictabilityDamped Oscillation: oscillations that become less pronounced as they

approach a stable point (e.g., carrying capacity)

Fluctuation: temporal changes in abundanceIrregular 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 equilibriumBifurcation: limit cycle that splits into n-periodicity

Chaos: bounded fluctuations with sensitive dependence on initial conditionsDeterministic 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 beingaffected 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

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

Characteristic Return Time, T

Single-Population Fluctuations:Mechanisms

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) )

Lets look at what happens to the predicted N with changes in population growth rate, R

N(t)

0

20

40

60

80

100

120

1 4 7 10 13 16 19 22 25 28 31

N(t)

Time

Pop

ula

tion

Siz

e

From Case (2000:115)

R < 2.0 Equilibrium point is locally stableR = 2.0 Equilibrium point becomes unstable

a cycle of 2 “is born”R = 2.449 A cycle of period 4 is bornR = 2.544 A cycle of period 8 is bornR = 2.564 A cycle of period 16 is bornR = 2.568 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.

Stable Limit Cycles

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

Single-Population Cycles:Mechanisms

Time delays cause oscillations and cycles in continuous models

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 arisesfrom 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

Observations and Hypotheses

Single-Population Cycles:Empirical Studies

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”