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Clone of Predator-Prey Model ("Lotka'Volterra")

Ethan Lee

Dynamic simulation modelers are particularly interested in understanding and being able to distinguish between the behavior of stocks and flows that result from internal interactions and those that result from external forces acting on a system.  For some time modelers have been particularly interested in internal interactions that result in stable oscillations in the absence of any external forces acting on a system.  The model in this last scenario was independently developed by Alfred Lotka (1924) and Vito Volterra (1926).  Lotka was interested in understanding internal dynamics that might explain oscillations in moth and butterfly populations and the parasitoids that attack them.  Volterra was interested in explaining an increase in coastal populations of predatory fish and a decrease in their prey that was observed during World War I when human fishing pressures on the predator species declined.  Both discovered that a relatively simple model is capable of producing the cyclical behaviors they observed.  Since that time, several researchers have been able to reproduce the modeling dynamics in simple experimental systems consisting of only predators and prey.  It is now generally recognized that the model world that Lotka and Volterra produced is too simple to explain the complexity of most and predator-prey dynamics in nature.  And yet, the model significantly advanced our understanding of the critical role of feedback in predator-prey interactions and in feeding relationships that result in community dynamics.The 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 3 months ago

MAT375 Clone of Bio103 Predator-Prey Model ("Lotka'Volterra")

Andrew E Long

Clone of Bio103 Predator-Prey Model ("Lotka'Volterra")
Tags: Education, Chaos, Ecology, Biology, PopulationThanks to Insight Author: John Petersen
Edits by Andy Long
Everything that follows the dashes was created by John Petersen (or at least came from his Insight model). I just wanted to make a few comments.
We are looking at Hare and Lynx, of course. Clone this insight, and change the names.

Then read the text below, to get acquainted with one of the most important and well-known examples of a simple system of differential equations in all of mathematics.

http://www.nku.edu/~longa/classes/mat375/mathematica/Lotka-Volterra.nb------------------------------------------------------------

Dynamic simulation modelers are particularly interested in understanding and being able to distinguish between the behavior of stocks and flows that result from internal interactions and those that result from external forces acting on a system. 

For some time modelers have been particularly interested in internal interactions that result in stable oscillations in the absence of any external forces acting on a system. 

The model in this last scenario was independently developed by Alfred Lotka (1924) and Vito Volterra (1926).  Lotka was interested in understanding internal dynamics that might explain oscillations in moth and butterfly populations and the parasitoids that attack them.  Volterra was interested in explaining an increase in coastal populations of predatory fish and a decrease in their prey that was observed during World War I when human fishing pressures on the predator species declined. 

Both discovered that a relatively simple model is capable of producing the cyclical behaviors they observed. 

Since that time, several researchers have been able to reproduce the modeling dynamics in simple experimental systems consisting of only predators and prey.  It is now generally recognized that the model world that Lotka and Volterra produced is too simple to explain the complexity of most predator-prey dynamics in nature.  And yet, the model significantly advanced our understanding of the critical role of feedback in predator-prey interactions and in feeding relationships that result in community dynamics.

The 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 Mat375 Lotka Volterra

  • 7 months 1 week ago

Prey&Predator

ilya
Физический смысл уравненийМодель Лотки-Вольтерры делает ряд предположений об окружающей среде и эволюции популяций хищников и жертв:
1. Хищная популяция всегда находит достаточно пищи.2. Продовольственная обеспеченность популяции хищника полностью зависит от размера популяции жертвы.3. Скорость изменения численности населения пропорциональна его численности.4. В ходе этого процесса окружающая среда не меняется в пользу одного вида, и генетическая адаптация не имеет существенного значения.5. Хищники обладают безграничным аппетитом.Поскольку используются дифференциальные уравнения, решение является детерминированным и непрерывным. Это, в свою очередь, означает, что поколения как хищника, так и жертвы постоянно пересекаются.
ДобычаКогда умножается, уравнение добычи становится
dx/dt = αx - βxy  Предполагается, что добыча имеет неограниченный запас пищи и размножается экспоненциально, если только она не подвержена хищничеству; этот экспоненциальный рост представлен в приведенном выше уравнении термином  αx. Предполагается, что скорость хищничества на добыче пропорциональна скорости, с которой встречаются хищники и добыча; это представлено выше в виде βxy.Если либо x, либо y равно нулю, то хищничества быть не может.С помощью этих двух терминов приведенное выше уравнение можно интерпретировать следующим образом: изменение численности добычи определяется ее собственным ростом минус скорость, с которой она охотится.ХищникиУравнение хищника становится

dy/dt =  - 

В этом уравнении,  представляет рост популяции хищника. (Обратите внимание на сходство со скоростью хищничества; однако используется другая константа, поскольку скорость роста популяции хищника не обязательно равна скорости, с которой он потребляет добычу).  представляет собой уровень потерь хищников вследствие естественной смерти или эмиграции; это приводит к экспоненциальному распаду в отсутствие добычи.


Следовательно, уравнение выражает изменение популяции хищников как рост, подпитываемый запасом пищи, минус естественная смерть.


Education Chaos Ecology Biology Population

  • 10 months 1 week ago

Clone of Population Stock and Flow

Alena Peskova
This is a simple population model designed to illustrate some of the concepts of stock and flow diagrams and simulation modelling.
The birth fraction and life expectancy are variables and are set as per page 66 of the text. The population is the stock and the births and deaths are the flows.

Education

  • 5 years 4 months ago

Clone of Predator-Prey Model ("Lotka'Volterra")

james gallagher

Dynamic simulation modelers are particularly interested in understanding and being able to distinguish between the behavior of stocks and flows that result from internal interactions and those that result from external forces acting on a system.  For some time modelers have been particularly interested in internal interactions that result in stable oscillations in the absence of any external forces acting on a system.  The model in this last scenario was independently developed by Alfred Lotka (1924) and Vito Volterra (1926).  Lotka was interested in understanding internal dynamics that might explain oscillations in moth and butterfly populations and the parasitoids that attack them.  Volterra was interested in explaining an increase in coastal populations of predatory fish and a decrease in their prey that was observed during World War I when human fishing pressures on the predator species declined.  Both discovered that a relatively simple model is capable of producing the cyclical behaviors they observed.  Since that time, several researchers have been able to reproduce the modeling dynamics in simple experimental systems consisting of only predators and prey.  It is now generally recognized that the model world that Lotka and Volterra produced is too simple to explain the complexity of most and predator-prey dynamics in nature.  And yet, the model significantly advanced our understanding of the critical role of feedback in predator-prey interactions and in feeding relationships that result in community dynamics.The 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

  • 2 years 8 months ago

Clone of Clone of Math Lab Routine Model

Edwin Gary Schasteen
This qualitative model will describe the potential daily routine of one professional mathematics tutor tutoring at a 2 year college, and should include all elements of the environment in which tutoring takes place, how policies affect the tutor, how the tutor's daily routine contributes to overall student success in key areas, and how those contributions are consistent, or inconsistent, with the stated mission and goals of the college. 
The model should include limitations/constraints that exist, and how those limitations effect other elements in the model.
Stage 1) Describe a skeleton model of the daily procedures a tutor would carry out on single day.
This current Model Contains a starting model of an individual human being in terms of generic Characteristics of majority of humans. This model contains elements like a brain, senses, ... etc. The brain part will be divided into functional brain regions including the limbic system, which has an impact on whether a particular behavior is reinforced and internalized, or suppressed.  The limbic system is likely  a major and relevant component in human learning, or the hindrance of student engagement with, and learning of, the material.  

Education

  • 5 years 2 months ago

Clone of Version 10: Hattie Calibrated Education Scenario Tool Capability Demonstration

Robert L. Brown
Despite a mature field of inquiry, frustrated educational policy makers face a crisis characterized by little to no clear research-based guidance and significant budget limitations --  in the face of too often marginal or unexpectedly deleterious achievement impacts. As such, education performance has been acknowledged as a complex system and a general call in the literature for causal models has been sounded. This modeling effort represents a strident first step in the development of an evidence-based causal hypothesis: an hypothesis that captures the widely acknowledged complex interactions and multitude of cited influencing factors. This non-piecemeal, causal, reflection of extant knowledge engages a neuro-cognitive definition of students.  Through capture of complex dynamics, it enables comparison of different mixes of interventions to estimate net academic achievement impact for the lifetime of a single cohort of students. Results nominally capture counter-intuitive unintended consequences: consequences that too often render policy interventions effete. Results are indexed on Hattie Effect Sizes, but rely on research identified causal mechanisms for effect propagation. Note that the net causal interactions have been effectively captured in a very scoped and/or simplified format.  Relative magnitudes of impact have been  roughly adjusted to Hattie Ranking Standards (calibration): a non-causal evidence source. This is a demonstration model and seeks to exemplify content that would be engaged in a full or sufficient model development effort.  Budget & time constraints required significant simplifying assumptions. These assumptions mitigate both the completeness & accuracy of the outputs. Features serve to symbolize & illustrate the value and benefits of causal modeling as a performance tool.

Education Policy Causal Dynamic Intervention Impact

  • 3 years 5 months ago

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