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Mathematics and BiologyMathematics and Biology
Biomathematics
eyond theeyond thesualsual
Brief Introduction toBrief Introduction to
MSP BiomathematicsInitiative
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Biomathematicsf Introduction tof Introduction to
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iomath as a fieldiomath as a fieldTheoretical Biology
omputational Biology
/Computational Biomodeling BiocomputingBioinformatics ,Creation of Algorithms Numerical Analysis and Simulation
ematical Biology
Biomathematics
iomodelinginding Solutions
nalysis
Experimental Biology
Biostatistics
Mathematical biophys
Systems Biology
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A LOOK INTO SYSTEMSBIOLOGY
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SIMULATIONIMULATION .OMP BIOOMP BIO
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EcologyEcologyPopulation dynamics
Malthus (1798):
Verhulst (1838, 1845):
(t)P0
P0P0
Biomath
/K 2
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EcologyEcologyInteraction of Species
Lotka-Volterra Predator-Prey Model:
Biomath
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EcologyEcologyPopulation Model with Age Structure
Leslie Matrix Model
( )LESLIE MATRIX
Biomath
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EpidemiologyEpidemiologyEpidemics
SIR Compartmental Model:
Id t
d RIIS
d t
d IIS
d t
d S === ,,
+ I + R = 1
I
Biomath
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ENZYME KINETICSENZYME KINETICS
-ich a e lis M e n te nn zy m e K in e tics
Biomath
h
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Mathematical PhysiologyMathematical Physiology
Biomath
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MATHs needed
Any branch of Mathematics can be; applied in Biology however
the field of biology should,dictate the maths needed not the.other way around
Biomathematics without biologistsBiomathematics without biologists.is too theoretical.is too theoretical
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Discrete MathDiscrete Math
Combinatorics (e.g. GenomeRearrangement)
Graph Theory and Network Analysis
Finite State Automata
Boolean Networks
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Knot Theory
AlgebraicStatistics(e.g.Phylogeny)
Fractals
o p o lo g y a n d M o d e rno p o lo g y a n d M o d e rnA lg e b ralg e b ra
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Optimization
EvolutionaryGame
Theory
DynamicProgrammin
g (e.g. DNASequenceAlignment)
p e ra tio n s R e se a rchp e ra tio n s R e se a rch
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Deterministic Models + Noise (e.g.diffusion)
Markov Chains
Bayesian Networks
Probabilistic Models (e.g. genetics)
;robability Stochasticrobability Stochasticrocess and Calculusrocess and Calculus
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Bifurcation and Chaos
Fib o n a cciS e q u e n c e
iffe re n ce E q u a tio niffe re n ce E q u a tio n
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Differential EquationsDifferential Equations
Every biomathematician should beEvery biomathematician should be
good in ODEs, PDEs and DDEs.good in ODEs, PDEs and DDEs.
Model and Analysis of DynamicalSystems
Control of Systems
MODEL PROCESSES AND CHANGE(cycles, switching, kinetics, rhythms,Input-Output, interaction,conduction, rate of growth/decayetc.)
/x dtx dt /y dty dt
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ODEsODEs
Solutions and Analysis of ODEs can bedone by:
1.Finding Explicit or Implicit Formula assolution
2.Finding series solution
3.Numerical analysis4.Representing solutions as graphs
5.Perturbation and stability analysis
6.Etc
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Perturbation & StabilityPerturbation & StabilityAnalysisAnalysis
- ( / /Fin d S te a d y sta te s E q u ilib riu m C ritica l R e st)Po in ts
- ,If S te a d y sta te is it sta b le o r u n sta b le ?Equilibrium Point
( )unstableEquilibrium Point
( )unstable
Equilibrium Point( )stable
Not in steady state
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Harvesting in CountvilleHarvesting in Countville
H a rv e stin g a S in g le Po p u la tio n w ith a co n sta n tyie ld Y 0 :
N
/dN dt
Y 0increasing
NFINITEECOVERYTIME
Y0= /rK 4
!Unstable
0
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Some Open ProblemsSome Open Problems(Chemostat Equations)(Chemostat Equations)
nn-species compete for-species compete for kkcomplementary resourcescomplementary resources
n=2 k=2 classic example n=3 k=2 competitive exclusion
n>3 k=2 open problem
n=4 k=3 co-existence may exist n=5 k=3 chaotic
n=12 k=5 coexists in oscillatory
form or chaotic form
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Romeo and JulietRomeo and Juliet
) , ( ) be Romeo s ardor for Juliet J t be Juliet s ardor for Rom
/ =dR dt aJ
/ =dJ dt bR
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Mathematics and BiologyMathematics and Biology
Bi th ti
eyond theeyond thesualsual
Brief Introduction toBrief Introduction to
MSP BiomathematicsInitiative