SLIDE 1
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
SLIDE 2 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
Phytoplankton Macrozooplankton Zooplankton
One gallon of water from the Chesapeake Bay can contain more than 500,000 zooplankton and
drop may contain thousands of individual phytoplankton!!
SLIDE 3 Why Study Plankton Population?
carbon fixation: CO2 concentration in atmosphere would be doubled without plankton indicators of nutrient level and other water quality conditions base
the food chain that support commercial fisheries plankton blooms that deplete oxygen
SLIDE 4 Population Model
∂P ∂t + u · ∇P = γP − ηP 2 + κ∇2P
advection small-scale diffusion growth saturation
✁ ✁ ✕ ✂ ✂✂ ✍ ❈ ❈ ❈ ❖ ❅ ❅ ■
SLIDE 5 Population Model
∂P ∂t + u · ∇P = γP − ηP 2 + κ∇2P
advection small-scale diffusion growth saturation
✁ ✁ ✕ ✂ ✂✂ ✍ ❈ ❈ ❈ ❖ ❅ ❅ ■
2D Lagrangian Chaotic Flow (U, k, τ)
x, t)= √ 2U cos[ky + α(t)]ˆ i , nτ ≤ t < (n+ 1
2)τ
√ 2U cos[kx + β(t)] ˆ j , (n+ 1
2)τ ≤ t < (n+1)τ
Spatially uniform γ and η: P → η/γ
SLIDE 6 Population Model
∂P ∂t + u · ∇P = γ( x)P − ηP 2 + κ∇2P
advection small-scale diffusion growth saturation
✁ ✁ ✕ ✂ ✂✂ ✍ ✄ ✄✄ ✗ ❇ ❇ ▼
2D Lagrangian Chaotic Flow (U, k, τ)
x, t)= √ 2U cos[ky + α(t)]ˆ i , nτ ≤ t < (n+ 1
2)τ
√ 2U 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)
SLIDE 7
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
SLIDE 8 An Example
doubly periodic domain domain size : 2π × 2π grid : 1024 × 1024
γ( x) = 0.2 + 0.8 cos x η = 1.0 κ ∼ 10−4 U = π k = 1 τ = 1/4
π 2π x
0.0 1.0 γ(x)
SLIDE 9 Biomass and Productivity
Biomass
B = P = lim
T→∞
1 TAΩ T dt
d x P( x, t)
Productivity
P = γP
For the above example: B ≈ 0.24 and P ≈ 0.14 (note γ = 0.2)
SLIDE 10 Biomass and Productivity
Biomass
B = P = lim
T→∞
1 TAΩ T dt
d x P( x, t)
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
SLIDE 11 Bounds on B and P
∂P ∂t + u · ∇P = γP − ηP 2 + κ∇2P (∗) (∗)
= 0 B = P + m
(m > 0) = m 4η
m 2 − mη
SLIDE 12 Bounds on B and P
∂P ∂t + u · ∇P = γP − ηP 2 + κ∇2P (∗) (∗)
= 0 B = P + m
(m > 0) ≤ m 4η
m 2 −mη
minimizing RHS with respect to m gives upper bound for B,
B ≤
2η
SLIDE 13 Bounds on B and P
(∗)/P γ − η P + κ
= 0 γ η ≤ B ≤
2η
SLIDE 14 Bounds on B and P
(∗)/P γ − η P + κ
= 0 γ η ≤ B ≤
2η
Cauchy-Schwarz Inequality: AB ≤
η P2 ≤ η
= γP ≤
γ2 η ≤ P ≤
η
SLIDE 15 Bounds on B and P
For our example with γ(
x) = 0.2 + 0.8 cos x and η = 1,
0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.0 0.1 0.2 0.3 0.4
P B
SLIDE 16 Simultaneous Bounds
P = γP + α[B − P] + β[γP − η
]
0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.0 0.1 0.2 0.3 0.4
f1(B) ≤ P ≤ f2(B) P B
SLIDE 17 What happen when γ < 0?
Recall γ
η ≤ B ≤
√
γ2+γ 2η
γ > 0 implies population will never become extinct
SLIDE 18 What happen when γ < 0?
Recall γ
η ≤ B ≤
√
γ2+γ 2η
γ > 0 implies population will never become extinct
When γ < 0, extinction (B = 0) is a possibility
π 2π
x
0.0 1.0
γ(x)
SLIDE 19
Survival-Extinction Transition
2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8
U
0.00 0.02 0.04 0.06
<P>
transition occurs at U = Uc ≈ 3.52, prediction for Uc?
SLIDE 20
Eddy (Effective) Diffusivity
Characterize spreading using X2: a pure diffusive process
∂φ ∂t = κ∇2φ ⇒ X2 = 4κt
pure advection by our flow model
∂φ ∂t + u · ∇φ = 0 ⇒ X2 = U 2τ 2 t
may parameterize the effect of
u on large-scale φ by D, ∂φ ∂t = D∇2φ
with
D = U 2τ 8
SLIDE 21 Theory for Uc
∂P ∂t + u · ∇P = γP − ηP 2 + κ∇2P
consider small κ near transition: ηP 2 ≈ 0 parameterization using eddy diffusivity D
∂P ∂t = γP + D∇2P
Linear stability analysis about P = 0: P = est ˆ
P
maximum s = 0 ⇒ extinction, thus critical Dc given by
∇2P + γ Dc P = 0
and
Dc = U 2
c T
8
SLIDE 22 Theory vs. Simulation
0.0
< γ >
2 3 4 5 6 7 8
Uc
theory simulation
SLIDE 23
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 Uc
