Variability in spaceIn time No migration migration (arithmetic) Source-sink structure with the rescue effect (geometric) G < A G declines with increasing variance Temporal variability reduces population growth rates Cure – populations decoupled with respect to variability, but coupled with respect to sharing individuals Source-sink structure (arith & geom) Increase the number of subpopulations increases the growth rate (to a point), and slows the time to extinction

Overview of population growth: discretecontinuous density independent density dependent Geometric Exponential Discrete Logistic New Concepts: - Stability - DI (non-regulating) vs. DD (regulating) growth - equilibrium Variability in growth (1)Individual variation in births and deaths (2)Environmental (extrinsic variability) (3)Intrinsic variability XX X

BUT, most populations appear more regulated than this….. And THERE ARE LIMITS TO GROWTH!!!! e.g., Australian sheep

Limits are manifested in (-) density dependence in population vital rates: mortality/survivorship reproduction

At higher densities, song sparrows: (a) smaller % reproductive males (b) fewer young fledged/female (c) lower juvenile survivorship Density dependence often affects more than a single component of those rates:

How do populations grow? time N Logistic Growth dN dt (K-N) K rN = 1 dN N dt (K-N) K r= population per capita N 1 dN N dt 0 K K K = Carrying capacity: the maximum density of individuals that the environment can support

If N = 0 (K-(0)) K = r 1 dN N dt (K-N) K r= K K = r

N If N = 0 (K-(0)) K = r 1 dN N dt (K-N) K r= K K = r That’s Exponential Growth } Exponential growth-like time

If N = K (K-(K)) K = r 1 dN N dt (K-N) K r= 0 K = r = 0

N K If N = K (K-(K)) K = r 1 dN N dt (K-N) K r= 0 K = r = 0 That’s Zero Growth } Zero growth time

N K 1 dN N dt (K-N) K r= Put the two together LOGISTIC GROWTH time

N 1 dN N dt 0 K 1 dN N dt (K-N) K r= r ( = r K_ K N K ) 1 _1 K )( N =r _ r K N Y = b + m X - growth + growth

2 nd Simplest expression of population growth: 2 parameters: r = intrinsic growth rate and K = carrying capacity Per capita growth rate is (-) density dependent Second Law of Ecology: There are limits to growth

N K time N EQ stability regulation Log.    Exp.

Rinderpest innoculation Severe drought

Rainfall Total food per capita food So what about Density-dependence?

Proportion of animals Live wildebeest Solid, white fat Opaque gelatinous Translucent gelatinous Lion/hyena killed Translucent gelatinous Opaque gelatinous Solid, white fat

http// To download a version of Populus:

r=0.2 r=1.0 r=1.8 r=2.0 Damped oscillations 2-point limit cycle Density time Discrete Logistic Growth

r=2.2 r=2.8r=4.0 r=2.5 Chaos 4-pt cycle extinction

r=2.8Chaos Chaos – “unpredictable” population dynamics incurred through very high growth rate and time lags between growth and negative feedback. Density time Extrinsic variability time

Density time K=1000; r=3.0

Islands < 1.0 ha support too few shrews to persist

Density time K=1000; r=3.0 Population culled by 25% time

Density Population culled by 25% Extrinsic variability

Variability comes in 2 flavors: Extrinsic and Intrinsic Recognizing the type of variability is important because different types require different solutions. Intrinsic –  growth rate or population size Extrinsic –  migration, # populations, population size

Overview of population growth: discretecontinuous density independent density dependent Geometric Exponential Discrete Logistic New Concepts: - Stability - DI (non-regulating) vs. DD (regulating) growth - equilibrium Variability in growth (1)Individual variation in births and deaths (2)Environmental (extrinsic variability) (3)Intrinsic variability XX X X XX

REVIEW - Populations consist of sources ( > 1) and sinks ( <1), the latter doom to extinction…….. - Populations have good years and bad years and temporal variation is bad …………………………… - Populations can grow chaotically by over- and under-shooting Carrying capacity…………………. - Populations with an Allee Effect can decline to extinction if N is too low……………………………….. - Cure: Dispersal from sources can Rescue sinks - Cure: Many populations that share individuals (dispersal) - Cull the population or otherwise reduce its growth - Recognize and keep density above the critical density

N time 2 Models of growth Exponential – all populations have the capacity to growth exponentially, but Growth has no limits and is density independent N 1 dN N dt 1 dN N dt = r Sustained Exponential growth is unrealistic

time N K 1 dN N dt (K-N) K r= Logistic – recognizes limits to growth (Carrying capacity) and incorporates the negative effect individuals have on their growth rate N 1 dN N dt 0 K r (- Density Dependence) Stable K

N 1 dN N dt 0 K One other variation is the ALLEE EFFECT where individuals also have + Density Dependence at low density + DD e.g., social behavior safety in numbers - DD Individuals inhibit their growth aahhhhh….

Important Concepts we have touch upon under Population Growth - Life Tables: Understanding how patterns of age-specific survivorship and maternity has consequences for population growth and can be manipulated to achieve a management goal - Variability: In space, populations exist as sources ( > 1) and sinks ( < 1), the latter of which must receive migrants to persist (Rescue Effect) In time, environmental variation is an anathema to population growth, but it too has a cure: increase the number of populations, migration, - Intrinsic Variability : Appreciate the difference between external and internal variation arising from time lags and delayed density dependence. Its cure is radically different than for external variation – and requires culling population size or otherwise reducing the growth rate.

Important Concepts we have touch upon under Population Growth - EQ, stability, and Pop. regulation: Attainable only under (-) density dependence. Negative feedback is Universal - Domains of Attraction: Specifically, under the Allee Effect, population extinction is an “attractant” below some critical density The concept of the limits to growth is manifested in the Carrying Capacity Species Social Behavior is manifested in the Allee Effect But otherwise, we have incorporated the biology of species as phenomena and have not appreciated the actual details But we will……

Where’s the Biology? Wildebeest populations growth competition for grass occurs Individuals are energy stressed Lions kill off weak individuals 1 dN N dt (K-N) K r= Lions? Grass? ??Energy/stress??

The Phenomenological Approach THE GOOD: Modeling the phenomena allows us to look past the details … we don’t need separate models for every organism THE BAD: We only get a superficial understanding …. when the details matter we’re left scratching our heads This tradeoff between DETAIL and GENERALITY Is pervasive throughout science