bistability in oyster differential equation model
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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
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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
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F
h
e
lif
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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
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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
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Figure 5: http://w
content/uploads/
df
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Logistic growth
•
Population growth•
= 1
• = ℎ
• =
Figure 6: http://education-portal.com/academy
population-growth-equation-definition-graph.h
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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
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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
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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
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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
8/11/2019 Bistability in Oyster Differential Equation Model
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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
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More equation
•
=
+ , =
2 • Proportion of oyster unaffected by sediment
• (, ) = − +
•
Modification that depends on reef height
• =
• Filtration rate per unit oyster volume
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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
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Bifurcation diagram for live adult oyster
Figure 12: Bifur
Matcont
Stable
unstable
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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
8/11/2019 Bistability in Oyster Differential Equation Model
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Larval functions
• = ×
+
• Larval production
• =
+
• Larval recruitment
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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
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Thank you!