Default tolerances are too large for Tsit5

[AshtonSBradley]

Describe the example

Predator-prey solution is wrong using defaults. Also occurs if we instead use defaults in DifferentialEquations.

Doesn't seem to occur if we switch to e.g. Vern6 defaults.

Minimal Reproducible Example 👇

    using OrdinaryDiffEq
    using Plots

    # system parameters
	λ = .7   # 🐇 linear growth - bunnies reproduce
	γ = 0.4  # 🐇 nonlinear loss - wolf eats rabbit
	κ = 0.3  # 🐺 nonlinear growth - wolf eats rabbit
	β = 1.2  # 🐺 linear loss - wolves get old
	p = [λ γ β κ]

    function pprey(u,p,t)
        λ,γ,β,κ = p 						
        r,w = u
        dr = λ*r - γ*r*w  		
        dw = - β*w + κ*r*w
        return [dr;dw]
    end 

r̄, w̄ = β/κ, λ/γ 	  
ti = 0.0 			
tf = 50.0 			
Nt = 200
t = LinRange(ti,tf,Nt)
alg = Tsit5()  
# alg = Vern6()		# Much better default

u0 = [β/κ + .2 ; λ/γ] # lots of carrots this month!  
prob = ODEProblem(pprey,u0,(ti,tf),p)

sol = solve(prob,alg=Tsit5(),saveat=t);  # default reltol=1e-3 (??!!)
# sol = solve(prob,alg=Tsit5(),saveat=t,reltol=1e-4); 
sol2 = solve(prob,alg=Vern6(),saveat=t);

r = sol[1,:]; w = sol[2,:];
r2= sol2[1,:]; w2 = sol2[2,:];

plot(r,w,size=(450,300),label="numerical solution")
plot!(r2,w2,label="correct numerical")
scatter!([r̄],[w̄],label="steady state",legend=:outerright) 
xlabel!("rabbits")
ylabel!("wolves")

Not Working Environment (please complete the following information):

• Output of using Pkg; Pkg.status()

julia> Pkg.status()
Status `~/Dropbox/Teaching/2024/phsi365/integrator_test/Project.toml`
⌃ [1dea7af3] OrdinaryDiffEq v6.49.4
  [91a5bcdd] Plots v1.40.2

• Output of using Pkg; Pkg.status(; mode = PKGMODE_MANIFEST)

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⌅ [79e6a3ab] Adapt v3.7.2
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⌃ [2b5f629d] DiffEqBase v6.130.0
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⌃ [1dea7af3] OrdinaryDiffEq v6.49.4
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• Output of versioninfo()

Julia Version 1.10.2
Commit bd47eca2c8a (2024-03-01 10:14 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: macOS (x86_64-apple-darwin22.4.0)
  CPU: 16 × Intel(R) Core(TM) i9-9900K CPU @ 3.60GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, skylake)
Threads: 8 default, 0 interactive, 4 GC (on 16 virtual cores)
Environment:
  JULIA_NUM_THREADS = 8
  JULIA_PKG_DEVDIR = /Users/abradley/Dropbox/Julia/Dev
  JULIA_EDITOR = code

Additional context

Here it is clear that this isn't the solution, which should be periodic, and is found by a small reduction in tolerance to e.g. reltol=1e-4. Alternately, switching to alg=Vern6() with default tolerances produces a credible solution.