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# Nonlinear ordinary differential equations | ||
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- lotka-volterra | ||
In this section we illustrate the flowpipe computation for a nonlinear system. | ||
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## Model description | ||
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Our running example is the [Lotka-Volterra model](https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations). The 2-dimensional Lotka-Volterra system depicts the populations change of a class of predators and a class of | ||
preys. The growth rate of preys’ population $x$ over time is given by | ||
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```math | ||
\dot{x} = x\cdot (\alpha - \beta \cdot y) | ||
``` | ||
wherein $\alpha, \beta$ are constant parameters and $y$ is the population of predators. | ||
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It gives that the number of preys grows exponentially without predation. | ||
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The population growth of predators is governed by the differential equation | ||
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```math | ||
\dot{y} = -y\cdot (\gamma - \delta\cdot x) | ||
``` | ||
wherein $\gamma, \delta$ are constant parameters. | ||
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We set those parameters as $\alpha = 1.5 , \beta = 1 , \gamma = 3$ and $\delta = 1$. | ||
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```@example lotka_volterra | ||
using ReachabilityAnalysis | ||
@taylorize function lotka_volterra!(du, u, p, t) | ||
local α, β, γ, δ = 1.5, 1.0, 3.0, 1.0 | ||
du[1] = u[1] * (α - β*u[2]) | ||
du[2] = -u[2] * (γ - δ*u[1]) | ||
return du | ||
end | ||
``` | ||
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## Reachability settings | ||
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The reachability settings are taken from [this resource](https://ths.rwth-aachen.de/research/projects/hypro/lotka-volterra/). | ||
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We consider the initial set $x\in [4.8,5.2], y \in [1.8,2.2]$. | ||
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```@example lotka_volterra | ||
X₀ = Hyperrectangle(low=[4.8, 1.8], high=[5.2, 2.2]) | ||
prob = @ivp(x' = lotka_volterra!(x), dim: 2, x(0) ∈ X₀) | ||
``` | ||
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## Results | ||
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We compute the flowpipe using the TMJets algorithm for the time horizon $[0,5]$: | ||
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```@example lotka_volterra | ||
sol = solve(prob, T=5.0) | ||
setrep(sol) | ||
``` | ||
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We can change to the zonotopic overapproximation of the flowpipe using | ||
the `overapproximate` function: | ||
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```@example lotka_volterra | ||
sol = overapproximate(sol, Zonotope) | ||
setrep(sol) | ||
``` | ||
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Finally we plot the solution in phase-space: | ||
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```@example lotka_volterra | ||
using Plots | ||
plot(sol, vars=(1, 2), xlab="x", ylab="y", lw=0.2, color=:lightblue, lab="Flowpipe") | ||
plot!(X₀, color=:orange, lab="Xo") | ||
``` |