Continuous Dynamical Systems Florence Hubert florence.hubert@univ-amu.fr Charlotte Perrin charlotte.perrin@univ-amu.fr Master CMB-BIO, 2019-2020
Practical information schedule ◮ lectures + exercises 12h ◮ computer sessions (language Python) 2 × 3h evaluation ◮ group project + reports on the computer sessions ◮ final exam
Mathematics and applications of mathematics http://mathsmonde.math.cnrs.fr https://images.math.cnrs.fr/
Basics in the modeling methodology Example of simple biological system: population of individuals unknown and variable ◮ the size of the population N is seen as a function of time: N = N ( t ) discrete description of the evolution of the system ◮ observation at discrete times t 0 = 0, t 1 = t 0 + ∆ t , t 2 = t 0 + 2∆ t , etc. ∆ t = constant time interval separating two observations ◮ variation of the population between two observations is given by N ( t i +1 ) − N ( t i ) = nb of births − nb of deaths ← conservation principle ◮ we prescribe how the nb of deaths and births (during ∆ t ) is related to N nb of births = λ ∆ t N ( t i ) , nb of deaths = µ ∆ t N ( t i ) ← comportemental law (Malthus) passage to the continuous description ∆ t → 0 d d t N ( t ) = aN ( t ) where a = λ − µ
Continuous dyn. systems � Differential Equations (ODEs) More generally ... (1st order) ODE problem: given a function f , find t �→ y ( t ) satisfying d � � d t y ( t ) = f t , y ( t ) ◮ when f only depends on y , the ODE is said to be autonomous ◮ when f depends linearly on y , the ODE is said to be linear Cauchy problem: given t 0 , y 0 (and f ), find y satisfying d � � d t y ( t ) = f t , y ( t ) y ( t 0 ) = y 0 ◮ y 0 is called the initial condition / initial datum
“Well-posed” problem J. Hadamard (1865-1963) → consistency of the models existence of a solution 1 uniqueness of the solution 2 continuous dependency with respect to the (initial) data (/ stability) 3 well-posedness of the Cauchy problem → Cauchy-Lipschitz Theorem
Notion of solution Cauchy problem: given t 0 ∈ I ⊂ R , y 0 ∈ U ⊂ R n , find y solution to d � � d t y ( t ) = f t , y ( t ) y ( t 0 ) = y 0 Definition (Notion of solution) A solution to the Cauchy problem is a couple ( J , y ) where J ⊂ I and y : J → U is a C 1 function satisfying the ODE for all t ∈ J. Definition (Global solution) A solution to the Cauchy problem is called global if J = I.
Example: solution to the Malthus model d d t N ( t ) = aN ( t ) N (0) = N 0 Linear, autonomous (1st order) ODE
Example: solution to the Malthus model d d t N ( t ) = a N ( t ) N (0) = N 0 Linear, autonomous (1st order) ODE global solution ( R , N ) given by N ( t ) = N 0 e at ∀ t ∈ R
Representation of the solution Cauchy problem: given t 0 ∈ I ⊂ R , y 0 ∈ U ⊂ R n , find y solution to d � � d t y ( t ) = f t , y ( t ) y ( t 0 ) = y 0 Definition (Notion of solution curves) Let ( J , Y ) a solution of the problem, we call solution curve of the system the set { ( t , Y ( t )) , t ∈ J } ⊂ R × R n . Definition (Notion of trajectories or orbits) Let ( J , Y ) a solution of the problem, we call trajectory or orbit of the system the set { Y ( t ) , t ∈ J } ⊂ R n .
Example: prey predator model � � x ′ ( t ) = f 1 � � x ( t ) , y ( t ) x (0) = x 0 y ′ ( t ) = f 2 � � x ( t ) , y ( t ) y (0) = y 0 number of preys x and predators y for different initial data left: the solution curves ( x in plain lines, y in dotted lines), right: the trajectories IV III II I
Some mathematical issues addressed in this course local (and global) well-posedness equilibrium points y ∗ which are such that f ( t , y ∗ ) = 0 for all t stability of equilibrium points long-time behavior other interesting issues: numerical discretization, dependency with respect to parameters and bifurcation phenomena, etc.
Outline ODEs and biology: most famous models linear systems of ODEs global well-posedness, stability of the origin, long-time behavior nonlinear ODEs conditions for local well-posedness, possible blow-up phenomena, equilibrium points and their stability
Some ODEs arising in biology Population dynamics Epidemiology Biochemistry Electrical networks in biology
Population dynamics: Malthusian growth model d d t N ( t ) = a N ( t ) N (0) = N 0 Linear, autonomous (1st order) ODE global solution ( R , N ) given by N ( t ) = N 0 e at ∀ t ∈ R exponential growth or decrease depending on the sign of a ref: Malthus (1798)
Malthusian catastrophe 1944: introduction of 29 reindeer on St Matthew island (Alaska) 1963: more than 6000 individuals observed (increase of more than 30% per year) 1966: almost the entire population is dead of starvation. ❀ Lichens had been completely eliminated ! ! the model does not take into account the possible limitation of the ressources ! A Malthusian catastrophe refers to a demographic collapse that follows an exponential growth of the population.
Models for one population with self-limiting growth → The Logistic model d Malthus: d t N ( t ) = a N ( t ) � 1 − N ( t ) � d d t N ( t ) = a N ( t ) � b b is the carrying capacity which depends on the environment non-linear 1st order (autonomous if a and b are constant) ODE equilibrium points: N ∗ 1 = 0, N ∗ 2 = b there exists a global solution and an analytical expression for N can be derived 30 Logistique, a=1, b=10 Logistique, a=1, b=20 Logistique, a=2, b=10 25 Logistique, a=2, b=20 N ( t ) → b as t → + ∞ 20 sigmoid shape of the sol. curves 15 10 ref: Verhulst (1838) 5 0 1 2 3 4 5
Models for one population with self-limiting growth → Gombertz and others.. Gombertz (1825): introduced for insurance companies, today in cancer modeling 30 Logistique, a=1, b=10 Logistique, a=1, b=20 Gompertz, a=1, b=10 25 Gompertz, a=1, b=20 � � b N ′ ( t ) = a ln 20 N ( t ) N ( t ) 15 ◮ analogue prop as the logistic model 10 5 0 2 4 6 8 10 more generally: N ′ ( t ) = h ( N ( t )) N ( t ) ◮ h ( N ) = N γ with γ < 0: power law model; ◮ h ( N ) = aN − 1 4 − b : West’s model; ◮ h ( N ) = aN − 1 3 − b : Von Bertalanffy’s model.
Models for several populations: the prey-predator model Biological assumptions in the absence of predators, the preys have an unlimited growth; the survival of the predators depends on the preys; the growth of the predator population is proportional to the size of the prey population. x : number of preys, y : number of predators, parameters a , b , c , d ≥ 0 � x ′ ( t ) = a x ( t ) − b x ( t ) y ( t ) y ′ ( t ) = d x ( t ) y ( t ) − c y ( t ) preys (water lilies) - predators (ducks)
� x ′ ( t ) = a x ( t ) − b x ( t ) y ( t ) y ′ ( t ) = d x ( t ) y ( t ) − c y ( t ) autonomous (the parameters are cst) system of two coupled non-linear ODEs no explicit solution IV III II I Left: solution curves, Right: corresponding trajectories in the xy-plane.
Extensions to more general predation rates Number of prey I consumed � x ′ ( t ) = ax ( t ) − bx ( t ) y ( t ) II III y ′ ( t ) = φ ( x ( t )) y ( t ) − cy ( t ) Density of prey population � � Holling Type I: φ ( x ) = d x 1 { x < ¯ x } + ¯ x 1 { x ≥ ¯ x } ◮ the time needed for the predators to find and consume preys is negligible dx Holling Type II: φ ( x ) = 1+ hdx ◮ h > 0: handling time of consumption ◮ limited capacity of the predator to find and eat the preys dx 2 Holling Type III: φ ( x ) = 1+ hdx 2 ◮ for large x , saturation effect similar to type II ◮ for small x , the predation rate remains low → preys can evade
Two populations in competition Two populations in the same environment exploiting the same limited resources Biological assumptions each population has a negative influence on the growth rate of the other each population satisfies a logistic growth in the absence of the other � 1 − x 1 ( t ) − α x 2 ( t ) � x ′ 1 ( t ) = a 1 x 1 ( t ) b 1 b 1 � � 1 − x 2 ( t ) − β x 1 ( t ) x ′ 2 ( t ) = a 2 x 2 ( t ) b 2 b 2 a 1 , 2 ≥ 0 are the intrinsic growth rates b 1 , 2 ≥ 0 are the carrying capacities α, β > 0 (resp. < 0) intensity of the competition (resp. symbiosis)
A model in epidemiology: the SIR model Time evolution of a population split into three parts population S of susceptible individuals; population I of infected individuals; population R of recovered (or immune) individuals. Biological assumptions the total population is constant; the average number of new infected individuals par unit time is proportional to the product of the infected and susceptible population sizes. S ′ ( t ) = − β S ( t ) I ( t ) + γ R ( t ) β : infection rate I ′ ( t ) = β S ( t ) I ( t ) − δ I ( t ) δ : recovery rate R ′ ( t ) = δ I ( t ) − γ R ( t ) γ : rate of transition between R and S Basic reproduction number R 0 If R 0 = β δ N > 1, the infection will be able to spread in the population. prediction of the spread of a disease possible extension to take into account the effect of a vaccination campaign...
Pharmacokinetics (PK)/Pharmacodynamics (PD) of a drug PK : How the organism affects the drug 1 PD : How the drug affects the organism 2
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