Examples of 2D Systems - Interactively

In the panel below you can explore paradigmatic dynamical systems, some of which exhibit multi-stability, periodic orbits or limit cycles. Some are typical activator-inhibitor systems, some are mutually inhibitory, others exhibit cooperative behavior. Below the interactive panel is a summary of each of the systems.

Quick system facts and summaries

Lotka Volterra

The Lotka-Volterra model captures the dynamics of a two species predator-prey system in which $x$ and $y$ represent the abundance of prey and predator, respectively: [ \dot{x} =x(\alpha-\beta y) ] [ \dot{y} =y(\gamma x-\delta) ]

Domain: The models domain is the region $x,y\geq0$.

Parameters: The system has four positive parameters, the prey reproduction rate $\alpha$, the rate $\beta$ at which prey descrease due to the predator, the predator reproduction rate $\gamma$ that is modulated by the abundance of prey, and rhe predator death rate $\delta$.

Fixpoints: The system has 2 stationary states

1. $(x^{\star},y^{\star})=(0,0)$
2. $(x^{\star},y^{\star})=(\delta/\gamma,\alpha/\beta)$

Null-clines:

• x: $x=0$ and $y=\alpha/\beta$
• y: $y=0$ and $x=\delta/\gamma$

Asympotitics: Periodic Solutions.

Comment: The Lotka-Volterra system is structurally unstable. Slight perturbations of the dynamical systems will qualitatively change the dynamics. Also, a predator-free system will yield exponential and unlimited growth of the prey.

Mutual inhibition

This dynamical system is characterized by two populations $x$ and $y$ that interact by mutually inhibitory forces onto each other captured by a growth rate that decreases with the abundance of the other. Plus there's some intra-population competitive regulation. [ \dot{x} =x(1-x-\alpha y) ] [ \dot{y} =y(1-y-\beta x) ]

Domain: The model's domain is the region $x,y\geq0$.

Parameters: The system has two non-negative parameters that capture the inhibition of $x$ on $y$ (parameter $\beta$) and $y$ on $x$ (parameter $\alpha$)

Fixpoints: The system always has three fixpoints

1. $(x^{\star},y^{\star})=(0,0)$
2. $(x^{\star},y^{\star})=(1,0)$
3. $(x^{\star},y^{\star})=(0,1)$

and if $\alpha,\beta<1$ or $\alpha,\beta>1$ also a fourth one

4. $(x^{\star},y^{\star})=(1-\alpha,1-\beta)/(1-\alpha\beta)$

Nullclines:

• x: $x=0$ and $y=1-\alpha y$
• y: $y=0$ and $y=1-\beta x$

Asympotitics: Depending on the parameter choices the system can have different attractors. If both inhibitory forces are weak, the coexisting state is the only attractor, if $\alpha>1$ and $\beta<0$ or vice versa only one population will prevail, the other going to zero. If both, $\alpha,\beta>0$ the sytem has two attractors and depending on the initial condition the system will approach one or the other.

Duffing Oscillator

The Duffing Oscillator is a mechanical system, a mass at position $x$ subject to a nonlinear force $F(x)=\alpha x-\beta x^{3}$ and a frictional force $-\gamma\dot{x}$ and the equation of motion: [ m\ddot{x}=\alpha x-\beta x^{3}-\gamma\dot{x} ] Letting $y=\dot{x}$, $m=1$, $\alpha,\beta,\gamma=1$ yields the dynamical system.

[ \dot{x} =y ] [ \dot{y} =-y+x-x^{3} ] Domain: The models domain is the entire $x-y-$plane.

Parameters: The original system has four parameters, the special case here has none.

Fixpoints: The system always has three fixpoints

1. $(x^{\star},y^{\star})=(0,0)$
2. $(x^{\star},y^{\star})=(1,0)$
3. $(x^{\star},y^{\star})=(0,1)$

The trivial fixpoint is an unstable saddle, the other two fixpoints are stable spirals Nullclines:

• x: $y=0$
• y: $y=x-x^{3}$

Asympotitics: Depending on the initial condition all trajectories will either spiral towards one or the other stable spiral fixpoints.

Comments: It is interesting to draw the basin of attraction for the Duffing-oscillator. Points that can be very close in the region of large $x$ and $y$ can move along almost the same trajectory but split because they are separated by the stable manifold of the fixpoint at the origin. Also, the duffing oscillator can behave in funky way when driven by a periodic external force.

SIRS epidemic model

This model captures the dynamics of an epidemic in which susceptibles (S) get infected by interacting with infecteds (I). Infecteds recover (R) and become immune. Recovered individuals remain immune for some time before they become susceptible again. All of this is captured by the reactions [ S+I\xrightarrow{\alpha}2I ] [ I\xrightarrow{\beta}R ] [ R\xrightarrow{\gamma}S ] denoting the fraction of susceptibles, infecteds and recovers by $x$,$y$ and $z$ yields the dynamical system [ \dot{x}=-\alpha xy+\beta(1-x-y) ] [ \dot{y}=\alpha xy-\beta y ] and $z=1-x-y$ because the number of individuals is conserved.

Domain: The models domain is $x,y>0$ and $x+y<1$.

Parameters: The model has three parameters, the recovery rate $\beta$ the transmission rate $\alpha$ and the rate of waning immuny $\gamma$. Typically it is assumed that $\alpha,\beta\gg\gamma$

Fixpoints: The system always has the fixpoint

1. $(x^{\star},y^{\star})=(1,0)$

which is stable if $\alpha<\beta$. If $\alpha>\beta$ the fixpoint

2. $(x^{\star},y^{\star})=(\beta/\alpha,(1-\beta/\alpha)/(1+\beta/\gamma))$

is a second fixpoint and is stable.

Nullclines:

• x: $y=\gamma(1-x)/(\gamma+\alpha x)$
• y: $y=0$ and $x=\beta/\alpha$

Asympotitics: If $\alpha/\beta>1$ all trajectories will approach the non-trivial stable fixpoint.

Predator Prey System

This system is a modification of the Lotka-Volterra system and, unlike it,structurally stable. The dynamical system is defined by [ \dot{x}=\alpha x(1-x)-\beta yx/(x+s) ] [ \dot{y}=\gamma yx/(x+s)-\delta y ] The modification compared to the LV system is that prey growth by itself is limited by a logistic, self-regulatory factor and that the predators feeding is saturated if prey are abundant beyond a concentration $s$.

Domain: The models domain is $x,y>0$.

Parameters: The model has 5 parameters. In addition to the four parameters of the Lotka-Volterra system, we have the additional saturation parameter $s$.

Fixpoints: The system always has the fixpoint

1. $(x^{\star},y^{\star})=(1,0)$

but depending on the parameters can also have the non-trivial fixpoint

2. $(x^{\star},y^{\star})=(\delta s/(\gamma-\delta),\alpha/\beta(\delta s/(\gamma-\delta)+s)(1-\delta s/(\gamma-\delta)))$

that is either stable or unstable, also depending on the parameter choice.

Nullclines:

• x: $x=0$ and $y=\alpha(x+s)(1-x)/\beta$
• y: $y=0$ and $x=\delta s/(\gamma-\delta)$

Asympotitics: For th right choice of parameters this system can have a limit cycle that attracts all trajectories, unlike the Lokta-Volterra system that has a different periodic orbit for differing intial conditions. For other parameter combinations, this system can have a stable non-trivial fixpoint.

Brusselator

This dynamical system with the odd name Brusselator is based on the chemical, autocatalytic reactions [ \emptyset\xrightarrow{\alpha}X ] [ 2X+Y\xrightarrow3X ] [ X\xrightarrow{\beta}Y ] [ X\xrightarrow\emptyset ] which yields the dynamical system [ \dot{x}=\alpha-(\beta+1)x+ x^{2}y ] and [ \dot{y}= x^{2}y-\beta x ]

Domain: The models domain is $x,y>0$.

Parameters: The model has 2 parameters for the two reactions that couple $X$ and $Y$ that are the core of the autocatalysis.

Fixpoints: The system has one fixpoint

1. $(x^{\star},y^{\star})=(\alpha,\beta/\alpha)$

depending on the parameter combination the fixpoint is stable or unstable.

Nullclines:

• x: $y=((\beta+1)x-\alpha)/ x^{2}$
• y: $x=0$ and $x=\beta/ s$

Asympotitics: Like the predator prey system above, the Brusselator can have a limit cycle, or an attractive spiral as a stable fixpoint.

Cooperation

This dynamical system is defined by the equations [ \dot{x}=R_{1}x(1-x)-x+\alpha xy(1-y) ] [ \dot{y}=R_{2}y(1-y)-y+\beta yx(1-x) ] for two populations. Without interaction ($\alpha,\beta=0$) each population grows logistically with an additional leak term (the second term in each ODE). The interaction is positive, abundance of $y$ will facilitate growth of $x$ and vice versa. This synergetic effect is also saturated by a logistic term.

Domain: The models domain is $x,y>0$.

Parameters: The model has 4 parameters, two parameters that control the reproduction of each population and two that control the cooperative influence of one on the other.

Fixpoints: This system can have a whole series of fixpoints, depending on the parameter combination. For some combinations up to 5 fixpoints. It is possible to compute them analytically but it is cumbersome.

Nullclines:

• x: $x=0$ and $y=1-1/R_{1}+\alpha x(1-x)/R_{1}$
• y: $x=y$ and $x=1-1/R_{2}+\beta y(1-y)/R_{2}$

Asympotitics: This system always has at least one stable fixpoint but can have up to three different stable fixpoints. This situation can be established with the reproduction parameters are low and the cooperativity parameters are high.