Bistability in Oyster Differential Equation Model

8/11/2019 Bistability in Oyster Differential Equation Model

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Bistability in oysterdifferential equation mod

Jing Yi Zhou

Advisers: Dr. Shaw, Dr. Shi, and Dr. Lipcius





Crassostrea virginica

•Filter feeder

• Spawns in early summer when

water temperatures rise

• Simultaneous hermaphrodites

• Adults release eggs and sperm into

the water.• Females can produce about 100

million eggs each year

Figure 1: http://sercblog.si.edu/?p=2842



F

h

e

lif



Oyster reefs

• Living reef

• Composed of clusters of oysters

• Rooted in a mixture of shells andshell fragments

• Cluster

• Colony of 3-7 generations of oysters

•Each generation becomes attachedto the shells of the preceding

Figure 3: Retrieved from 20

Lipcius oyster restoration p



Current status

•

Ermgassen compared results of fishery-independent surveyChesapeake oysters from about 100 years ago to results of msurveys in 2012.

• Remaining biomass ranged from 3% to 38% in various regioChesapeake Bay.

•

This is mainly attributed to overfishing and diseases (Examp





Figure 5: http://w

content/uploads/

df



Logistic growth

•

Population growth•

= 1

• = ℎ

• =

Figure 6: http://education-portal.com/academy

population-growth-equation-definition-graph.h



Equilibria or steady-states

•

It is a solution of an ODE that does notchange with time

• Can be classified as stable or unstable bylooking at the signs of the eigenvalue of theJacobian matrix

• Example)

=

2 1

• Equilibria located where

= 0

at P = 0, 10

Figure 7:

http://tutorial.math.lamar.edu/C

spx



Bifurcations

•

A small change in the parameter can leadto a drastic change in the long-termbehavior of solutions. Such a change iscalled a bifurcation.

• Types of bifurcations

• Saddle-node (fold) bifurcation

Figure 8: Retrieved from paper. Regime shifts

can occur with no

Warning by Hastings, Ecology Letters, 2010, v



Bistability or alternative stable states

•

Existence of multiple equilibriagiven a range of constants

• Example: Catastrophic bifurcation

Figure 9: Retrieved from paper. Regime shif

can occur with no

Warning by Hastings, Ecology Letters, 2010



What causes these alternative stable sta

• Usually due to one or more feedback mechanisms

• For example)

• Low sediment: Oysters filter sediment → lowers turbidit

→ increases dissolved oxygen concentration → more oys

• High sediment: Oysters filter sediment → oysters spend

→ increases susceptibility to disease → decreases growt

reproduction



Previous differential equations

•

= 1

1

•

= 1

•

=

−

Natural

mortality

Death from

sedimentOyster growth rate

Shells from dead oysters Degradation of shell

Oyster

filtration rate

Sediment

erosion



More equation

•

=

+ , =

2 • Proportion of oyster unaffected by sediment

• (, ) = − +

•

Modification that depends on reef height

• =

• Filtration rate per unit oyster volume



Times series

•

At the initial value (O, B, S) =(0.01, 0.39, 0.01)

Figure 9: Retrieved from paper. Jordan-Cooley, Bistability

in a differential equation model of oyster reef height and

sediment accumulation

Figure 10: Time series generated through Matlab





Bifurcation diagram for live adult oyster

Figure 12: Bifur

Matcont

Stable

unstable



Current model

•

=

•

= ()

1

•

= 1

•

=

−

Maturation of

oyster juveniles

Natural

mortality

Death from

sedimentOyster growth rate

Shells from dead oysters Degradation of shell

Oyster

filtration rate

Sediment

erosion

Larvae recruitment and

production



Larval functions

• = ×

+

• Larval production

• =

+

• Larval recruitment



Objectives

• Finding all the necessary parameters

• Parameters conversion

• Defining parameters that are biologically plausible

• Do bifurcation analysis on the new model

• Couple populations that interact



Thank you!

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Bistability in Oyster Differential Equation Model