Plankton Model with Time Delayed Nutrient Recycling Sue Ann - PowerPoint PPT Presentation

Plankton Model with Time Delayed Nutrient Recycling Sue Ann Campbell, Matthew Kloosterman and Francis Poulin Department of Applied Mathematics University of Waterloo Southern Ontario Dynamics Day April 12, 2013

Outline Introduction/Background 1 Existence of Equilibrium Points 2 Stability of Equilibrium Points 3 No Delay With Delay Conclusions and Implications 4

Introduction Plankton are free floating organisms found in oceans and lakes which form the bottom of the food chain.

Introduction Phytoplankton are plankton which carry out photosynthesis examples: diatoms, golden algae, green algae and cyanobacteria

Introduction Zooplankton are plankton that feed on phytoplankton examples: jelly fish, small crustaceans and insect larvae

Motivation Why study plankton? Plankton form the bottom of the ocean food chain.

Motivation Why study plankton? Plankton form the bottom of the ocean food chain. Phytoplankton can exhibit blooms which can be harmful to ecosystem and humans.

Motivation Why study plankton? Plankton form the bottom of the ocean food chain. Phytoplankton can exhibit blooms which can be harmful to ecosystem and humans. Phytoplankton are very important in the transfer of carbon dioxide from the atmosphere to the ocean.

Model Closed model with three compartments: dissolved nutrient - N ( t ) phytoplankton - P ( t ) zooplankton - Z ( t ) (measured by amount of limiting nutrient/nitrogen) P .J.S. Franks (2002) J Oceanogr. 58:379-387.

Model Closed model with three compartments: dissolved nutrient - N ( t ) phytoplankton - P ( t ) zooplankton - Z ( t ) (measured by amount of limiting nutrient/nitrogen) Phytoplankton Uptake Grazing Recycling Nutrient Zooplankton Recycling P .J.S. Franks (2002) J Oceanogr. 58:379-387.

Model Model with three compartments: dissolved nutrient - N ( t ) phytoplankton - P ( t ) zooplankton - Z ( t ) (measured by amount of limiting nutrient/nitrogen) N ′ ( t ) = λ P ( t ) + δ Z ( t ) + ( 1 − γ ) gZ ( t ) h ( P ( t )) − µ P ( t ) f ( N ( t )) P ′ ( t ) = µ P ( t ) f ( N ( t )) − gZ ( t ) h ( P ( t )) − λ P ( t ) Z ′ ( t ) = γ gZ ( t ) h ( P ( t )) − δ Z ( t )

Model Model with three compartments: dissolved nutrient - N ( t ) phytoplankton - P ( t ) zooplankton - Z ( t ) (measured by amount of limiting nutrient/nitrogen) N ′ ( t ) = λ P ( t ) + δ Z ( t ) + ( 1 − γ ) gZ ( t ) h ( P ( t )) − µ P ( t ) f ( N ( t )) P ′ ( t ) = µ P ( t ) f ( N ( t )) − gZ ( t ) h ( P ( t )) − λ P ( t ) Z ′ ( t ) = γ gZ ( t ) h ( P ( t )) − δ Z ( t ) phytoplankton nutrient uptake

Model Model with three compartments: dissolved nutrient - N ( t ) phytoplankton - P ( t ) zooplankton - Z ( t ) (measured by amount of limiting nutrient/nitrogen) N ′ ( t ) = λ P ( t ) + δ Z ( t ) + ( 1 − γ ) gZ ( t ) h ( P ( t )) − µ P ( t ) f ( N ( t )) P ′ ( t ) = µ P ( t ) f ( N ( t )) − gZ ( t ) h ( P ( t )) − λ P ( t ) Z ′ ( t ) = γ gZ ( t ) h ( P ( t )) − δ Z ( t ) zooplankton grazing on phytoplankton

Model Model with three compartments: dissolved nutrient - N ( t ) phytoplankton - P ( t ) zooplankton - Z ( t ) (measured by amount of limiting nutrient/nitrogen) N ′ ( t ) = λ P ( t ) + δ Z ( t ) + ( 1 − γ ) gZ ( t ) h ( P ( t )) − µ P ( t ) f ( N ( t )) P ′ ( t ) = µ P ( t ) f ( N ( t )) − gZ ( t ) h ( P ( t )) − λ P ( t ) Z ′ ( t ) = γ gZ ( t ) h ( P ( t )) − δ Z ( t ) zooplankton grazing on phytoplankton nutrient recycling

Model Model with three compartments: dissolved nutrient - N ( t ) phytoplankton - P ( t ) zooplankton - Z ( t ) (measured by amount of limiting nutrient/nitrogen) N ′ ( t ) = λ P ( t ) + δ Z ( t ) + ( 1 − γ ) gZ ( t ) h ( P ( t )) − µ P ( t ) f ( N ( t )) P ′ ( t ) = µ P ( t ) f ( N ( t )) − gZ ( t ) h ( P ( t )) − λ P ( t ) Z ′ ( t ) = γ gZ ( t ) h ( P ( t )) − δ Z ( t ) zooplankton and phytoplankton death

Model Model with three compartments: dissolved nutrient - N ( t ) phytoplankton - P ( t ) zooplankton - Z ( t ) (measured by amount of limiting nutrient/nitrogen) N ′ ( t ) = λ P ( t ) + δ Z ( t ) + ( 1 − γ ) gZ ( t ) h ( P ( t )) − µ P ( t ) f ( N ( t )) P ′ ( t ) = µ P ( t ) f ( N ( t )) − gZ ( t ) h ( P ( t )) − λ P ( t ) Z ′ ( t ) = γ gZ ( t ) h ( P ( t )) − δ Z ( t ) zooplankton and phytoplankton death nutrient recycling

Model Parameters Parameter Meaning Units day − 1 µ phytoplankton maximum growth rate day − 1 λ phytoplankton death rate day − 1 g zooplankton maximum grazing rate γ zooplankton assimilation efficiency day − 1 δ zooplankton death rate N ′ ( t ) = λ P ( t ) + δ Z ( t ) + ( 1 − γ ) gZ ( t ) h ( P ( t )) − µ P ( t ) f ( N ( t )) P ′ ( t ) = µ P ( t ) f ( N ( t )) − gZ ( t ) h ( P ( t )) − λ P ( t ) Z ′ ( t ) = γ gZ ( t ) h ( P ( t )) − δ Z ( t )

Functional Response Nutrient uptake by phytoplankton: µ P ( t ) f ( N ( t )) f ( 0 ) = 0 , f ′ ( N ) ≥ 0 , f ′′ ( N ) ≤ 0 , lim N →∞ f ( N ) = 1 (Michaelis-Menten/Type II) W.C. Gentleman & A.B. Neuheimer (2008) J. Plankton Research 30(11) 1215-1231.

Functional Response Nutrient uptake by phytoplankton: µ P ( t ) f ( N ( t )) f ( 0 ) = 0 , f ′ ( N ) ≥ 0 , f ′′ ( N ) ≤ 0 , lim N →∞ f ( N ) = 1 (Michaelis-Menten/Type II) Zooplankton grazing on phytoplankton: gZ ( t ) h ( P ( t )) h ( 0 ) = 0 , h ′ ( P ) ≥ 0 , lim P →∞ h ( P ) = 1 (Type II or III) W.C. Gentleman & A.B. Neuheimer (2008) J. Plankton Research 30(11) 1215-1231.

Functional Response Nutrient uptake by phytoplankton: µ P ( t ) f ( N ( t )) f ( 0 ) = 0 , f ′ ( N ) ≥ 0 , f ′′ ( N ) ≤ 0 , lim N →∞ f ( N ) = 1 (Michaelis-Menten/Type II) Zooplankton grazing on phytoplankton: gZ ( t ) h ( P ( t )) h ( 0 ) = 0 , h ′ ( P ) ≥ 0 , lim P →∞ h ( P ) = 1 (Type II or III) W.C. Gentleman & A.B. Neuheimer (2008) J. Plankton Research 30(11) 1215-1231.

Model N ′ ( t ) = λ P ( t ) + δ Z ( t ) + ( 1 − γ ) gZ ( t ) h ( P ( t )) − µ P ( t ) f ( N ( t )) P ′ ( t ) = µ P ( t ) f ( N ( t )) − gZ ( t ) h ( P ( t )) − λ P ( t ) Z ′ ( t ) = γ gZ ( t ) h ( P ( t )) − δ Z ( t ) Phytoplankton Uptake Grazing Recycling Nutrient Zooplankton Recycling

Model N ′ ( t ) = λ P ( t ) + δ Z ( t ) + ( 1 − γ ) gZ ( t ) h ( P ( t )) − µ P ( t ) f ( N ( t )) P ′ ( t ) = µ P ( t ) f ( N ( t )) − gZ ( t ) h ( P ( t )) − λ P ( t ) Z ′ ( t ) = γ gZ ( t ) h ( P ( t )) − δ Z ( t ) Include distributed time delay in recycling

Model N ′ ( t ) = λ P ( t ) + δ Z ( t ) + ( 1 − γ ) gZ ( t ) h ( P ( t )) − µ P ( t ) f ( N ( t )) P ′ ( t ) = µ P ( t ) f ( N ( t )) − gZ ( t ) h ( P ( t )) − λ P ( t ) Z ′ ( t ) = γ gZ ( t ) h ( P ( t )) − δ Z ( t ) Include distributed time delay in recycling � ∞ N ′ ( t ) = [ λ P ( t − u ) + δ Z ( t − u ) + ( 1 − γ ) gZ ( t − u ) h ( P ( t − u ))] η ( u ) du 0 − µ P ( t ) f ( N ( t )) P ′ ( t ) = µ P ( t ) f ( N ( t )) − gZ ( t ) h ( P ( t )) − λ P ( t ) Z ′ ( t ) = γ gZ ( t ) h ( P ( t )) − δ Z ( t ) � ∞ � ∞ where η ( u ) du = 1 , τ = u η ( u ) du (mean delay) 0 0

Model N ′ ( t ) = λ P ( t ) + δ Z ( t ) + ( 1 − γ ) gZ ( t ) h ( P ( t )) − µ P ( t ) f ( N ( t )) P ′ ( t ) = µ P ( t ) f ( N ( t )) − gZ ( t ) h ( P ( t )) − λ P ( t ) Z ′ ( t ) = γ gZ ( t ) h ( P ( t )) − δ Z ( t ) Include distributed time delay in recycling � ∞ N ′ ( t ) = [ λ P ( t − u ) + δ Z ( t − u ) + ( 1 − γ ) gZ ( t − u ) h ( P ( t − u ))] η ( u ) du 0 − µ P ( t ) f ( N ( t )) P ′ ( t ) = µ P ( t ) f ( N ( t )) − gZ ( t ) h ( P ( t )) − λ P ( t ) Z ′ ( t ) = γ gZ ( t ) h ( P ( t )) − δ Z ( t ) � ∞ � ∞ where η ( u ) du = 1 , τ = u η ( u ) du (mean delay) 0 0 Recycling time is u ∈ [ 0 , ∞ ) with probability η ( u ) .

Distributions Gamma distribution: η ( u ) = u p − 1 � p � p e − pu /τ τ Γ( p ) � 1 2 W , τ − W ≤ u ≤ τ + W Uniform distribution: η ( u ) = , 0 , elsewhere  u + W − τ , τ − W ≤ u ≤ τ  W 2 − u + W + τ Tent distribution: η ( u ) = , τ ≤ u ≤ τ + W . W 2  0 , elsewhere Discrete delay: η ( u ) = δ ( u − τ )

Distributions ( τ = 2) 0.6 0.5 0.4 g 0.3 0.2 0.1 0.0 0 1 2 3 4 5 u Gamma ( p = 1 , 2 , 4 , 8) Uniform ( W = 0 . 5 , 1 , 2) Tent ( W = 0 . 5 , 1 , 2)

Conservation Laws Model with no delay: N ′ ( t ) = λ P ( t ) + δ Z ( t ) + ( 1 − γ ) gZ ( t ) h ( P ( t )) − µ P ( t ) f ( N ( t )) P ′ ( t ) = µ P ( t ) f ( N ( t )) − gZ ( t ) h ( P ( t )) − λ P ( t ) Z ′ ( t ) = γ gZ ( t ) h ( P ( t )) − δ Z ( t )

Conservation Laws Model with no delay: N ′ ( t ) = λ P ( t ) + δ Z ( t ) + ( 1 − γ ) gZ ( t ) h ( P ( t )) − µ P ( t ) f ( N ( t )) P ′ ( t ) = µ P ( t ) f ( N ( t )) − gZ ( t ) h ( P ( t )) − λ P ( t ) Z ′ ( t ) = γ gZ ( t ) h ( P ( t )) − δ Z ( t ) Total nutrient in system is conserved. N ( t ) + P ( t ) + Z ( t ) = N T (constant)

Conservation Laws Model with delay: � ∞ N ′ ( t ) = [ λ P ( t − u ) + δ Z ( t − u ) + ( 1 − γ ) gZ ( t − u ) h ( P ( t − u ))] η ( u ) du 0 − µ P ( t ) f ( N ( t )) P ′ ( t ) = µ P ( t ) f ( N ( t )) − gZ ( t ) h ( P ( t )) − λ P ( t ) Z ′ ( t ) = γ gZ ( t ) h ( P ( t )) − δ Z ( t )