Mortality over Time Population Density Declines through Mortality

Experimental Evidence: Self Thinning Log mean plant weight (w ) Log density (N) LowHigh Low High Change during one time interval

Experimental Evidence: Self Thinning Log mean plant weight (w ) Log density (N) LowHigh Low High Change during one time interval

Experimental Evidence: Self Thinning Log mean plant weight (w ) Log density (N) LowHigh Low High Change during one time interval

Experimental Evidence: Self Thinning Log mean plant weight (w ) Log density (N) LowHigh Low High Change during one time interval

Experimental Evidence: Self Thinning Log mean plant weight (w ) Log density (N) General pattern 1.Unimpeded growth 2.Mortality begins 3.Similar trajectories exhibited once thinning starts 4.At some point thinning slows

Self Thinning in Thirty Species Similar slope to thinning line across a range of species

Attempts to Explain the Thinning Line

An Intuitive Argument Two stands of trees starting at different densities

An Intuitive Argument Two stands of trees starting at different densities Thinning occurs as trees increase in size.

An Intuitive Argument Two stands of trees starting at different densities Thinning occurs as trees increase in size. Trees cannot grow larger unless enough space is made available through mortality.

Yoda et al. (1963) propose the “-3/2 Thinning Law” k ≈ -3/2

“-3/2 Thinning” k ≈ -3/2 Allometric relationships: those that scale with body mass They posit an underlying allometric relationship

“-3/2 Thinning” k ≈ -3/2 They posit an underlying allometric relationship w = average individual biomass C = constant N = population density -k = slope of thinning line

“-3/2 Thinning” k ≈ -3/2 They posit an underlying allometric relationship Why 3/2?

An Intuitive Argument

Biomass Density  Volume–> m 3  Area  m2  m2

An Intuitive Argument Biomass Density  Volume–> m 3  Area  m2  m2

An Intuitive Argument Biomass Density  Volume–> m 3  Area  m2  m2

An Intuitive Argument Biomass Density  Volume–> m 3  Area  m2  m2

An Intuitive Argument Biomass Density  Volume–> m 3  Area  m2  m2

k ≈ -3/2k ≈ -4/3 Revisiting the “-3/2 Thinning Law” X

k ≈ -3/2k ≈ -4/3 A Revised View of the Allometric Relationship Same as the scaling relationship of body mass to maximum density in animals!

A General Interpretation of the Thinning Relationship

Lemna Sequoia A General Interpretation of the Thinning Relationship

Permitted combinations Prohibited combinations

Self Thinning Revisited Log mean plant weight (w ) Log density (N) General pattern 1.Unimpeded growth 2.Mortality begins 3.Similar trajectories exhibited once thinning starts 4.At some point thinning slows 4 ?

Self Thinning Revisited Log mean plant weight (w ) Log density (N) Growth limited by space Growth limited by resources

Self Thinning Revisited Log mean plant weight (w ) Log density (N) Growth limited by resources Resource limitation regulating growth leads to the “Law of Constant Yield”

Proof of Constant Yield with a slope = -1 Log mean plant weight Log density Slope ≈ -1 log(N)log(N-z) log Y N Y (N-z)

Proof of Constant Yield with a slope = -1 Log mean plant weight Log density Slope ≈ -1 log(N)log(N-z) log Y N Y (N-z) Calculation of slope

Proof of Constant Yield with a slope = -1 Log mean plant weight Log density log(N)log(N-z) log Y N Y (N-z) Calculation of slope XX

Proof of Constant Yield with a slope = -1 Log mean plant weight Log density log(N)log(N-z) log Y N Y (N-z) Calculation of slope = -1

Putting it all together Development of size hierarchies Thinning Law of Constant Yield