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wolf ~ boom and bust

K Phu
This model is to be used with Mr. Roderick's AP biology activity on population growth. See for a copy of the activity worksheet.
Use the sliders below to quickly change the initial values of components of the model.

Ecology Computer Modeling

  • 4 years 6 months ago

A More Realistic Model of Isle Royale: Predator Prey Interactions

Andrew E Long
This model illustrates predator prey interactions using real-life data of wolf and moose populations on the Isle Royale.

We incorporate logistic growth into the moose dynamics, and we replace the death flow of the moose with a kill rate modeled from the kill rate data found on the Isle Royale website.

A decent match to the data is made with
Wolf Death Rate = 0.15
Wolf Birth Rate Factor = 0.0203
Moose Death Rate Factor = 1.08
Moose Birth Rate = 0.4
Carrying Capacity = 2000
Initial Moose: 563
Initial Wolves: 20

I used RK-4 with step-size 0.1, from 1959 for 60 years.

The moose birth flow is MBR*M*(1-M/K)
Moose death flow is MDRF*Sqrt(M*W)
Wolf birth flow is WBRF*Sqrt(M*W)
Wolf death flow is WDR*W

Environment Ecology Populations Midterm

  • 1 year 8 months ago

Isle Royale: Predator/Prey Model for Moose and Wolves, with Total Population

Andrew E Long
This model illustrates predator prey interactions using real-life data of wolf and moose populations on the Isle Royale. It was "cloned" from a model that InsightMaker provides to its users, at
Thanks Scott Fortmann-Roe.

I've added in an adjustment to handle population.

I've created a Mathematica file that replicates the model, at

It allows one to experiment with adjusting the initial number of moose and wolves on the island.

I used steepest descent in Mathematica to optimize the parameters, with my objective data being the ratio of wolves to moose. You can try my (admittedly) kludgy code, at

woStart} =

Environment Ecology Populations Math Modeling

  • 1 year 8 months ago

Prey&Predator - 3z MA

Johan Raunkjær Borre
​Physical meaning of the equationsThe Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations:
1. The prey population finds ample food at all times.2. The food supply of the predator population depends entirely on the size of the prey population.3. The rate of change of population is proportional to its size.4. During the process, the environment does not change in favour of one species and genetic adaptation is inconsequential.5. Predators have limitless appetite.As differential equations are used, the solution is deterministic and continuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping.[23]
When multiplied out, the prey equation becomesdx/dt = αx - βxy The prey are assumed to have an unlimited food supply, and to reproduce exponentially unless subject to predation; this exponential growth is represented in the equation above by the term αx. The rate of predation upon the prey is assumed to be proportional to the rate at which the predators and the prey meet; this is represented above by βxy. If either x or y is zero then there can be no predation.

With these two terms the equation above can be interpreted as: the change in the prey's numbers is given by its own growth minus the rate at which it is preyed upon.


The predator equation becomes

dy/dt =  - 

In this equation, {\displaystyle \displaystyle \delta xy} represents the growth of the predator population. (Note the similarity to the predation rate; however, a different constant is used as the rate at which the predator population grows is not necessarily equal to the rate at which it consumes the prey). {\displaystyle \displaystyle \gamma y} represents the loss rate of the predators due to either natural death or emigration; it leads to an exponential decay in the absence of prey.

Hence the equation expresses the change in the predator population as growth fueled by the food supply, minus natural death.

Education Chaos Ecology Biology Population

  • 1 month 1 week ago

Selection under logistic population dynamics

Barry McMullin
Logistic population dynamics with selection: Two species, able to exploit one resource, which is available at  a fixed, finite, flow (not a depleting stock). At low populations, growth is exponential. As long as total population below total carrying capacity, growth in total population continues, stabilizing at carrying capacity. Similar stabilization "from above" (if, e.g., carrying capacity dynamically falls to some lower level). "Smooth" stabilization (formally logistic) in either case. But selection between species if respective carrying capacities are different (regardless of intrinsic growth rates?); or, if equal carrying capacities, then selection based on intrinsic growth rates (but much slower?).


  • 1 year 3 months ago

Dystopia: simple energy system model

Barry McMullin
Basic idea is to model demand with endogenous growth (but "satiation" becomes possible - eventually - at some notional "sufficiency" level); and supply then attempts to track demand with some time lag (~5-50 years - characteristic of commissioning/decommissioning large scale energy infrastructure). But supply also produces pollution, which accoumulates. We can specify a notional constraint/limit; approaching this should trumpdemand and forces supply to zero. In this version we'll only have one source. so no substitution is possible. We expect to see a fairly sudden supply crash. Of course, "demand" will still carry merrily on its way up anyway, but the interpretation of the consequently growing supply shortfall will be left to the eye of the beholder. NB: this version doesn't automatically succeed in limiting P to P_max. It forces dS/dt to zero as A*(P/ P_max) reaches 1; and then as that value exceeds 1, dS/dt is forced negative. But this dynamics has no way to "undo" any overshoot of P over P_max (which would require S itself to become negative: "negative emissions"). Need to manual find/choose a big enough value of A to limit P effectively.

Energy Environment Ecology

  • 1 year 3 months ago