Planktonic Population in a Spatially Variable Environment Yue-Kin - PowerPoint PPT Presentation

Planktonic Population in a Spatially Variable Environment Yue-Kin Tsang Courant Institute of Mathematical Sciences New York University Daniel Birch and William R. Young Scripps Institution of Oceanography, UCSD

What is Plankton? tiny open-water bacteria, plants or animals that have limited or no swimming ability transported through the water by currents and tides One gallon of water from the Chesapeake Bay Phytoplankton can contain more than 500,000 zooplankton and Zooplankton one drop may contain thousands of individual phytoplankton!! Macrozooplankton

Why Study Plankton Population? carbon fixation: CO 2 concentration in atmosphere would be doubled without plankton indicators of nutrient level and other water quality conditions base of the food chain that support commercial fisheries plankton blooms that deplete oxygen

Population Model ∂P u · ∇ P = γP − ηP 2 + κ ∇ 2 P ∂t + � ✁ ✕ ■ ❅ ❈ ❖ ✂✂ ✍ ✁ ❅ ❈ advection small-scale diffusion ✂ ❈ growth saturation

Population Model ∂P u · ∇ P = γP − ηP 2 + κ ∇ 2 P ∂t + � ✁ ✕ ❖ ❈ ■ ❅ ✂✂ ✍ ✁ ❅ ❈ advection small-scale diffusion ✂ ❈ growth saturation 2D Lagrangian Chaotic Flow ( U , k , τ ) √ 2 U cos[ ky + α ( t )]ˆ  nτ ≤ t < ( n + 1 i , 2 ) τ  � u ( � x, t )= √ 2 U cos[ kx + β ( t )] ˆ j , ( n + 1 2 ) τ ≤ t < ( n +1) τ  Spatially uniform γ and η : P → η/γ

Population Model ∂P x ) P − ηP 2 + κ ∇ 2 P ∂t + � u · ∇ P = γ ( � ✁ ✕ ❇ ▼ ✄✄ ✗ ✂✂ ✍ ✁ ❇ advection small-scale diffusion ✂ ✄ growth saturation 2D Lagrangian Chaotic Flow ( U , k , τ ) √ 2 U cos[ ky + α ( t )]ˆ  nτ ≤ t < ( n + 1 i , 2 ) τ  � u ( � x, t )= √ 2 U cos[ kx + β ( t )] ˆ j , ( n + 1 2 ) τ ≤ t < ( n +1) τ  Spatially uniform γ and η : P → η/γ Environmental variability : γ = γ ( � x )

Oasis and Desert interested in the situation where γ ( � x ) > 0 in some region (oasis) and γ ( � x ) ≤ 0 in the complementary region (desert) γ ≤ 0 γ > 0 two cases: (1) � γ � > 0 and (2) � γ � < 0

An Example 1.0 doubly periodic domain γ (x) domain size : 2 π × 2 π 0.0 grid : 1024 × 1024 -0.6 π 0 2 π x γ ( � x ) = 0 . 2 + 0 . 8 cos x η = 1 . 0 κ ∼ 10 − 4 U = π k = 1 τ = 1 / 4

Biomass and Productivity Biomass � T 1 � B = � P � = lim dt d� x P ( � x, t ) TA Ω T →∞ 0 Ω Productivity P = � γP � For the above example: B ≈ 0 . 24 and P ≈ 0 . 14 (note � γ � = 0 . 2 )

Biomass and Productivity Biomass � T 1 � B = � P � = lim dt d� x P ( � x, t ) TA Ω T →∞ 0 Ω Productivity P = � γP � For the above example: B ≈ 0 . 24 and P ≈ 0 . 14 (note � γ � = 0 . 2 ) We shall obtain bounds on B and P in terms of γ and η

Bounds on B and P ∂P u · ∇ P = γP − ηP 2 + κ ∇ 2 P ∂t + � ( ∗ ) γP − ηP 2 � � � ( ∗ ) � = 0 γP − ηP 2 � � B = � P � + m ( m > 0) �� � 2 � = m γ + 1 ( P − #) 2 � � − mη 4 η m

Bounds on B and P ∂P u · ∇ P = γP − ηP 2 + κ ∇ 2 P ∂t + � ( ∗ ) γP − ηP 2 � � � ( ∗ ) � = 0 γP − ηP 2 � � B = � P � + m ( m > 0) �� � 2 � ≤ m γ + 1 ( P − #) 2 � � − mη 4 η m minimizing RHS with respect to m gives upper bound for B , � � γ 2 � + � γ � B ≤ 2 η

Bounds on B and P |∇ ln P | 2 � � � ( ∗ ) /P � γ − η � P � + κ = 0 � � γ 2 � + � γ � � γ � ≤ B ≤ η 2 η

Bounds on B and P |∇ ln P | 2 � � � ( ∗ ) /P � γ − η � P � + κ = 0 � � γ 2 � + � γ � � γ � ≤ B ≤ η 2 η � � A 2 � � B 2 � , Cauchy-Schwarz Inequality: � AB � ≤ η � P � 2 ≤ η � P 2 � � � γ 2 � � P 2 � = � γP � ≤ � γ � 2 � γ 2 � ≤ P ≤ η η

Bounds on B and P For our example with γ ( � x ) = 0 . 2 + 0 . 8 cos x and η = 1 , 0.4 0.3 0.2 P 0.1 0.0 0.20 0.30 0.40 0.15 0.25 0.35 0.45 B

Simultaneous Bounds � P 2 � P = � γP � + α [ B − � P � ] + β [ � γP � − η ] f 1 ( B ) ≤ P ≤ f 2 ( B ) 0.4 0.3 0.2 P 0.1 0.0 0.20 0.30 0.40 0.15 0.25 0.35 0.45 B

What happen when � γ � < 0 ? √ � γ 2 � + � γ � Recall � γ � η ≤ B ≤ 2 η � γ � > 0 implies population will never become extinct

What happen when � γ � < 0 ? √ � γ 2 � + � γ � Recall � γ � η ≤ B ≤ 2 η � γ � > 0 implies population will never become extinct When � γ � < 0 , extinction ( B = 0 ) is a possibility 1.0 γ (x) 0.0 -0.5 π 0 2 π x

Survival-Extinction Transition 0.06 0.04 <P> 0.02 0.00 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 U transition occurs at U = U c ≈ 3 . 52 , prediction for U c ?

Eddy (Effective) Diffusivity Characterize spreading using X 2 : a pure diffusive process ∂φ X 2 = 4 κt ∂t = κ ∇ 2 φ ⇒ pure advection by our flow model X 2 = U 2 τ ∂φ ∂t + � u · ∇ φ = 0 ⇒ 2 t may parameterize the effect of � u on large-scale φ by D , D = U 2 τ ∂φ ∂t = D ∇ 2 φ with 8

Theory for U c ∂P u · ∇ P = γP − ηP 2 + κ ∇ 2 P ∂t + � consider small κ near transition: ηP 2 ≈ 0 parameterization using eddy diffusivity D ∂P ∂t = γP + D ∇ 2 P Linear stability analysis about P = 0 : P = e st ˆ P maximum s = 0 ⇒ extinction, thus critical D c given by D c = U 2 ∇ 2 P + γ c T P = 0 and D c 8

Theory vs. Simulation 8 7 theory simulation 6 5 U c 4 3 2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 < γ >

Summary we study plankton population using an advection-diffusion logistic-growth model in a domain divided into oasis ( γ > 0 ) and desert ( γ < 0 ) for � γ � > 0 , population never extinct and we obtain bounds on the biomass and productivity for � γ � < 0 , population becomes extinct when the velocity is large, we make prediction to such critical velocity U c