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Ecology

Rachel Driehaus Midterm MAT 375

Rachel Driehaus
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.

I start with these parameters:
Wolf Death Rate = 0.15
Wolf Birth Rate = 0.0187963
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 logistic, MBR*M*(1-M/K)
Moose death flow is Kill Rate (in Moose/Year)
Wolf birth flow is WBR*Kill Rate (in Wolves/Year)
Wolf death flow is WDR*W

Environment Ecology Populations Midterm

  • 3 years 3 months ago

Clone of Jacob Englert MAT 375 Midterm: Model of Isle Royale: Predator Prey Interactions

Urasa Kongdechakul
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.

I start with these parameters:
Wolf Death Rate = 0.15
Wolf Birth Rate = 0.0187963
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 logistic, MBR*M*(1-M/K)
Moose death flow is Kill Rate (in Moose/Year)
Wolf birth flow is WBR*Kill Rate (in Wolves/Year)
Wolf death flow is WDR*W

Environment Ecology Populations Midterm

  • 1 year 11 months ago

Logistic stabilization of population

Barry McMullin
Logistic population dynamics: One 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 population below carrying capacity, growth 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.

Ecology

  • 2 years 10 months ago

Dystopia-1: 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 tracks demand with some time lag (~5-50 years(?), characteristic of commissioning/decommissioning large scale energy infrastructure). Then add cumulative pollution, with a hard constraint/limit which trumps demand and forces supply (of any non-zero polluting source) to zero. In this version we have one source (so no substitution is possible), and it produces a cumulative pollutant, so  we expect to see supply decline and/or crash (according to the specific parameters and dynamics). 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. In this version we try to "smooth" the decline - using the fractional "exhaustion" of the pollution "quota" as feedback signal to smoothly shift from a dynamic of "supply chasing demand" and one of "exponential mitigation of supply within the remaining pollution quota". This particular dynamic is rigid about not exceeding the quota: it does not allow (and could not cope with) overshoot. There is also no provision for delay in the feedback (that could perhaps be added, and would presumably allow a more prolonged addiction, but then more rapid and painful withdrawal?).

Energy Environment Ecology

  • 2 years 10 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]
Prey
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.

Predators

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 year 8 months ago

Lotka-Volterra Model: Prey-Predator Simulation

Pavan Kumar Guntur

​Predator-prey models are the building masses of the bio-and environments as bio masses are become out of their asset masses. Species contend, advance and scatter essentially to look for assets to support their battle for their very presence. Contingent upon their particular settings of uses, they can take the types of asset resource-consumer, plant-herbivore, parasite-have, tumor cells- immune structure, vulnerable irresistible collaborations, and so on. They manage the general misfortune win connections and thus may have applications outside of biological systems. At the point when focused connections are painstakingly inspected, they are regularly in actuality a few types of predator-prey communication in simulation. 

 

Looking at Lotka-Volterra Model:

The well known Italian mathematician Vito Volterra proposed a differential condition model to clarify the watched increment in predator fish in the Adriatic Sea during World War I. Simultaneously in the United States, the conditions contemplated by Volterra were determined freely by Alfred Lotka (1925) to portray a theoretical synthetic response wherein the concoction fixations waver. The Lotka-Volterra model is the least complex model of predator-prey communications. It depends on direct per capita development rates, which are composed as f=b−py and g=rx−d. 

A detailed explanation of the parameters:

  • The parameter b is the development rate of species x (the prey) without communication with species y (the predators). Prey numbers are reduced by these collaborations: The per capita development rate diminishes (here directly) with expanding y, conceivably getting to be negative. 

  • The parameter p estimates the effect of predation on x˙/x. 

  • The parameter d is the death rate of species y without connection with species x. 

  • The term rx means the net rate of development of the predator population in light of the size of the prey population.

Reference:

http://www.scholarpedia.org/article/Predator-prey_model

 

Education Chaos Ecology Biology Population

  • 1 year 10 months 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?).

Ecology

  • 2 years 10 months ago

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