Hallo!

Carol Horvitz Professor of Biology University of Miami, Florida, USA plant population biology, spatial and temporal

variation in demography applications to plant-animal interactions, invasion

biology, global change, evolution of life span

Institute for Theoretical and Mathematical Ecology

University of MiamiCoral Gables, FL USA

Mathematics

Steve CantrellChris CosnerShigui Ruan

BiologyDon De AngelisCarol HorvitzMatthew Potts

Marine ScienceJerry AultDon Olson

Dynamics of structured populations

N(t+1) = N(t) * pop growth rate Pop growth rate depends upon

Survival and reproduction of individuals

Survival, growth and reproduction are not uniform across all individuals

Thus the population is structured

Population dynamics: changes in size and shape of populations

Demographic structure age stage size space year habitat

Modeling dynamics life table matrix life cycle graph

Age vs. stage?

Regression Log-linear

Projection

n(t+1) = A n(t)

Population projection matrix

Stage attimet+1

Stage at time t

seed seedling juvenile reproductive

seed 0.1

seedling 0.2

juvenile 0

reproductive

0

Population projection matrix

Stage attimet+1

Stage at time t

seed seedling juvenile reproductive

seed 0

seedling 0.1

juvenile 0.3

reproductive

0.1

Population projection matrix

Stage attimet+1

Stage at time t

seed seedling juvenile reproductive

seed 0

seedling 0

juvenile 0.1

reproductive

0.2

Population projection matrix

Stage attimet+1

Stage at time t

seed seedling juvenile reproductive

seed 12

seedling 0

juvenile 0

reproductive

0.4

Population projection matrix

Stage attimet+1

Stage at time t

seed seedling juvenile reproductive

seed 0.1 0 0 12

seedling 0.2 0.1 0 0

juvenile 0 0.3 0.1 0

reproductive

0 0.1 0.2 0.4

Life cycle graph

try it

Start with 10 in each stage class multiply and add row times column

Population projection matrix

0.1 0 0 12

0.2 0.1 0 0

0 0.3 0.1 0

0 0.1 0.2 0.4

10

10

10

10

try it

Start with 10 in each stage class

Start with 72, 17, 6 and 5 in the stage classes

Population projection matrix

0.1 0 0 12

0.2 0.1 0 0

0 0.3 0.1 0

0 0.1 0.2 0.4

try it

Start with 10 in each stage class n(2) = 121, 3, 4, 7 Start with 72, 17, 6 and 5 in the stage classes n(2) = 67,16, 6, 5 population growth rate = 0.9564

Projection

n(1) = A n(0)n(2) = A n(1)n(3) = A n(2)n(4) = A n(3)

n(5) = A n(4)n(6) = A n(5)

time

Projection

n(t+1) = A n(t)

Projection

n(1)= A n(0)n(2)= AAn(0)n(3)= AAAn(0)n(4)= AAAAn(0)n(5)= AAAAAn(0)n(6)=AAAAAAn(0)

Projection

n(t) = At n(0)

Projection

n(t+1) = A n(t)Each time step, the population changes size and shape.

The matrix pulls the population into different shapes.

There are some shapes that are ‘ in tune ’ with the environment.

For these, the matrix only acts to change the size of the population.

In these cases the matrix acts like a scalar.

Projection

n(t+1) = A n(t)

n(t+1) = n(t)

Projection

n(t+1) = A n(t)Examples: stable stage

reproductive valuessensitivity to perturbation

time variantdensity dependentother

Projection exercises

Stable age distribution and population growth rate

Reproductive value of different ages

Not all matrices yield a stable age distribution concentration of

reproduction in the last age

oscillations

Analytical entities

Dominant eigenvalue Dominant right eigenvector (ssd) Dominant left eigenvector (rv) Derivative of population growth rate with

respect to each element in the matrix Derivative of the logarithm of population

growth rate with respect to the logarithm of each element in the matrix