1 birth rate γ (number of births per time interval) 2 death rate δ proportional to size of population

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1 Scienific Comuing I Module : Poulaion Modelling Coninuous Models Michael Bader Par I ODE Models Lehrsuhl Informaik V Winer 7/ Discree vs. Coniuous Models d d = F,,...) ) =? discree model: coninuous model: ) N individuals : R R,) =? Move o Coninuous Models: easier?) ye of mahemaical roblem: differenial euaions, calculus analyical soluions available?) Model of Malhus Differenial Euaion Wrien as an ordinary differenial euaion: ṗ) = r ) Reuires iniial condiion oulaion a sar) ) = Model of Malhus 79) Only one secies: birh rae γ number of birhs er ime inerval) roorional o size of oulaion deah rae δ roorional o size of oulaion hus: consan growh or decay) rae: r = γ δ Modelling: consan growh rae d d = r growh wihin a ime inerval + δ) = ) + r ) δ Model of Malhus Soluions The model of Malhus describes exonenial growh or decay of a oulaion:, Analyical soluion: ) = e r,,

2 Model of Verhuls 9h cenury) Objecive: model oulaions ha aroach sauraion value Assumions: growh/deah rae deend on oulaion size; assume linear deendency: g) = g g ) d) = d + d ) Model of Verhuls Sauraion solve iniial value roblem: ṗ) = α β), ) = soluion: ) = + e β ), = α β, leads o differenial euaion:, ṗ) = g) d) = g d ) g }{{} + d ) ) }{{} =:α =:β, Model of Verhuls Logisic Growh sauraion model does no longer model exonenial growh idea: le growh/deah rae decrease linearly wih size of oulaion bu kee growh/deah rae roorional o oulaion size leads o differenial euaion: ṗ) = α β))) Logisic Growh oher formulaion soluion:,, ṗ) = α ) ) ) β β ) = e α ) + β e α, Logisic Growh wih Threshold exended version of Verhuls s model: ṗ) = α ) ) ) ) ) β δ soluions β =,δ = ): Examle The Passenger Pigeon beginning of he 9h cenury, esimaed oulaion in Norh America: four billion huning diminished is number below a criical hreshold lae s) The las assenger igeon died on Seember, s 9.

3 Scienific Comuing I Module : Poulaion Modelling Coninuous Models Pars II and III) Michael Bader Par II More Than One Secies Sysems of ODE Lehrsuhl Informaik V Winer 7/ A Linear Model Firs Examle: Arms Race similar o Verhuls s sauraion model addiional growh erm roorional o oher secies leads o yically: ṗ) = b + a ) + a ) ) = b + a ) + a ) b >,b > growh erm) a <,a < sauraion) a,a? armamen of wo hosile) counries our susicion: a >, a > Observaion: long-ime behaviour deends on size of arameers seady-sae soluions exis soluions exis ha show unlimied growh Second Examle: Comeiion A Non-Linear Model wo secies sharing a common naural habia comeiion: a <, a < Observaion: long-ime behaviour deends on size of arameers seady-sae soluions exis some scenarios are hysically incorrec! negaive oulaion size) similar o Verhuls s logisic growh model addiional growh erm roorional o oher secies leads o yically: ṗ) = b + a ) + a ))) ) = b + a ) + a ))) b >,b > growh erm) a <,a < sauraion) a,a?

4 The Non-Linear Comeiion Model wo secies sharing a common naural habia comeiion: a <, a < Possible Scenarios: seady-sae one secies dies ou exincion) no obvious nonsense Comeiion Seady Sae ṗ) = + ) )) ) ) = 7 + ) )) 7) soluion for =, = : Comeiion Exincion ṗ) = ) = 7 7 soluion for =, = : ) ) )) ) ) )) Predaor-Prey wo secies: redaor and rey redaor eas rey: a > rey is eaen by redaor: a < Possible Scenarios: sable oscillaions one secies dies ou wha haens wih he oher, hen?) Predaor-Prey by Loka & Volerra ṗ) = + )) ) ) = )) ) soluion for =, = : Oen Quesions Mehods o Analyse a Given Model? redic aroximae soluion or shae of soluion? redic ossible seady saes? redic criical oins? secies on edge of exicion?) Mehods o Imrove Modeling? redic failure of he model? une arameers o model a secific siuaion?

5 Analysing he Sloe of a Soluion Par III Discussion and Analysis of ODE Models Examle: Model of Malhus ṗ) = α) for a sensible soluion: ) > α decides sloe of soluion: α > : growing oulaion acceleraed growh) α < : receding oulaion deceleraed reducion) Poins of Euilibrium Examle: Model of Verhuls sauraion) euilibrium: ṗ) = only, if ) = α β ṗ) = α β) Examle: Logisic Growh ṗ) = α ) ) ) β Criical Poins Observaion on Logisic Growh: consan soluion ) = β, if ) = β consan soluion ) =, if ) = euilibrium a = β is reached for nearly all iniial condiions aracive sable) euilibrium euilibrium a = is no reached for any oher iniial condiions reulsive ) unsable euilibrium consan soluion, if ) = β or ) = Criical Poins Derivaives Examine derivaives: criical oin = aracive euilibrium asymoically sable): unsable euilibrium: ṗ < for = + ε ṗ > for = ε Direcion Field lo derivaives vs. ime and size of oulaion: Examle: Logisic Growh ṗ) = α ) ) ) β, ) oherwise: saddle oin ṗ > for = + ε ṗ < for = ε,,

6 Direcion Field ) Examle: Logisic Growh wih Threshold ṗ) = α ) ) ) ) ) β δ Idenifying Criical Poins aracive euilibrium: ) unsable euilibrium saddle oin Criical Poins in D Examle: Arms Race sysem of differenial euaions euilibrium: ṗ =, = ṗ) = b + a ) + a ) = ) = b + a ) + a ) = soluion of a linear sysem of euaions: a ) + a ) = b a ) + a ) = b in mos cases one criical oin criical line, if sysem marix is singular D Direcion Field Arms Race ṗ) = ) + ) ) = + ) ) Direcion Field for a Sysem of ODE examle: D ṗ) = b + a ) + a ) ) = b + a ) + a ) naural exension: D lo: vs. vs. D direcion field for vs. or vs. no sufficien: wha values o chose for or res.)? bu: saionary roblem indeenden of hus: lo direcions deending on and Arms Race unlimied growh ṗ) = ) + ) ) = + ) ) direcion field wih criical oin a,): direcion field wih criical oin a,):,,

7 Arms race he eaceful neighbour ṗ) = ) + ) ) = ) ) direcion field wih criical oin a, ) : Nonlinear Sysem Comeiion ṗ) = + ) )) ) ) = 7 + ) )) 7) direcion field criical oins a,),...:,,,,,, Nonlinear Sysem Exincion ṗ) = ) = 7 7 ) ) )) ) ) )) criical oins a,.7...),...,),...: Loka & Volerra ṗ) = + )) ) ) = )) ) direcion field wih criical oin a, ):,,, D Criical Poins Summary Differen yes of criical oins in D: aracive/sable euilibrium arms race seady sae) unsable euilibrium saddle oin arms race unlimied growh) aracive siral oin eaceful neighbour ) unsable siral oin cenre of roaion Loka-Volerra) How o discriminae beween hese yes? Homogeneous Sysems of ODE Homogeneous Sysem in marix-vecor-noaion: x : R R n, A R n n ẋ = Ax examle: x) = ), )) Soluions: le x λ be an eigenvecor: Ax λ = λx λ hen x λ e λ is a soluion: Ax λ e λ = λx λ e λ = d d x λ e λ).e.d.

8 Eigenvecors and Eigenvalues Corollaries: he soluions of he homogeneous sysem ẋ = Ax are linear combinaions of he resecive eigen-soluions: x hom ) = a λ x λ e λ, λ a λ R he soluions of he inhomogeneous sysem ẋ = Ax + b are x) = A b + x hom ) observaion: x c = A b is a criical oin! Eigenvalues and Criical Poins he ODE sysem ẋ = Ax + b is solved by x c aracive euilibrium, x) = x c + a λ x λ e λ λ lim x) = x c, only if e λ for all eigenvalues λ λ R λ < λ = µ + iν µ < e iν = cosν + i sinν) Sabiliy of Linear Sysems Overview: eigenval. λ j = µ j + iν j ) criical oin sabiliy Sabiliy of D Sysems Real Eigenvalues: λ <, λ <, aracive euilibrium eig x eig real, all λ < node sable, ar. real, all λ > node unsable real, λ k >,λ l < saddle oin unsable comlex, all µ < siral oin sable, ar. comlex, all µ > siral oin unsable comlex, all µ = cenre sable x Sabiliy of D Sysems Real Eigenvalues: λ >, λ >, unsable euilibrium Sabiliy of D Sysems Real Eigenvalues: λ >, λ <, saddle oin eig x eig eig x eig x x

9 Sabiliy of D Sysems Comlex Eigenvalues: µ <, µ <, siral oin asym. sable) Sabiliy of D Sysems Comlex Eigenvalues: µ >, µ >, siral oin unsable) eig x eig eig x eig x x Sabiliy of D Sysems Sabiliy of Non-Linear Sysems Comlex Eigenvalues: µ = µ =, cenre of oscillaion eig x eig D sysem of ODE: ẋ) = fx)), f : R n R n nonlinear criical oin a x c : fx c ) = for analysis of criical oins: linearizaion x ẋ) = fx)) fx c ) +J }{{} f x c )x) x c ) = examine eigenvalues of J f x c )