Population Models
These models and simulations have been tagged “Population”.
Related tagsEcologyEnvironmentBiologyChaosEducationDemographics
Month Basis of Lake Sturgeon Population Model
Simulation of MTBF with controls
The inverse curve is the trust time
On the right the increase in failures brings its inverse which is loss of trust and move into suspicion and lack of confidence.
This can be seen in strategic social applications with those who put economy before providing the priorities of the basic living infrastructures for all.
This applies to policies and strategic decisions as well as physical equipment.
A) Equipment wears out through friction and preventive maintenance can increase the useful lifetime,
B) Policies/working practices/guidelines have to be updated to reflect changes in the external environment and eventually be replaced when for instance a population rises too large (constitutional changes are required to keep pace with evolution, e.g. the concepts of the ancient Greeks, 3000 years ago, who based their thoughts on a small population cannot be applied in 2013 except where populations can be contained into productive working communities with balanced profit and loss centers to ensure sustainability)
Early Life
Useful Life
Wearout
F(t) = 1 - e ^ -λt
Where
• F(t) is the probability of failure
• λ is the failure rate in 1/time unit (1/h, for example)
• t is the observed service life (h, for example)
The inverse curve is the trust time
On the right the increase in failures brings its inverse which is loss of trust and move into suspicion and lack of confidence.
This can be seen in strategic social applications with those who put economy before providing the priorities of the basic living infrastructures for all.
This applies to policies and strategic decisions as well as physical equipment.
A) Equipment wears out through friction and preventive maintenance can increase the useful lifetime,
B) Policies/working practices/guidelines have to be updated to reflect changes in the external environment and eventually be replaced when for instance a population rises too large (constitutional changes are required to keep pace with evolution, e.g. the concepts of the ancient Greeks, 3000 years ago, who based their thoughts on a small population cannot be applied in 2013 except where populations can be contained into productive working communities with balanced profit and loss centers to ensure sustainability)
Early Life
If we follow the slope from the leftmost start to where it begins to flatten out this can be considered the first period. The first period is characterized by a decreasing failure rate. It is what occurs during the “early life” of a population of units. The weaker units fail leaving a population that is more rigorous.
Useful Life
The next period is the flat bottom portion of the graph. It is called the “useful life” period. Failures occur more in a random sequence during this time. It is difficult to predict which failure mode will occur, but the rate of failures is predictable. Notice the constant slope.
Wearout
The third period begins at the point where the slope begins to increase and extends to the rightmost end of the graph. This is what happens when units become old and begin to fail at an increasing rate. It is called the “wearout” period.
Clone of BATHTUB MEAN TIME BETWEEN FAILURE (MTBF) RISK
Simulation of MTBF with controls
The inverse curve is the trust time
On the right the increase in failures brings its inverse which is loss of trust and move into suspicion and lack of confidence.
This can be seen in strategic social applications with those who put economy before providing the priorities of the basic living infrastructures for all.
This applies to policies and strategic decisions as well as physical equipment.
A) Equipment wears out through friction and preventive maintenance can increase the useful lifetime,
B) Policies/working practices/guidelines have to be updated to reflect changes in the external environment and eventually be replaced when for instance a population rises too large (constitutional changes are required to keep pace with evolution, e.g. the concepts of the ancient Greeks, 3000 years ago, who based their thoughts on a small population cannot be applied in 2013 except where populations can be contained into productive working communities with balanced profit and loss centers to ensure sustainability)
Early Life
Useful Life
Wearout
F(t) = 1 - e ^ -λt
Where
• F(t) is the probability of failure
• λ is the failure rate in 1/time unit (1/h, for example)
• t is the observed service life (h, for example)
The inverse curve is the trust time
On the right the increase in failures brings its inverse which is loss of trust and move into suspicion and lack of confidence.
This can be seen in strategic social applications with those who put economy before providing the priorities of the basic living infrastructures for all.
This applies to policies and strategic decisions as well as physical equipment.
A) Equipment wears out through friction and preventive maintenance can increase the useful lifetime,
B) Policies/working practices/guidelines have to be updated to reflect changes in the external environment and eventually be replaced when for instance a population rises too large (constitutional changes are required to keep pace with evolution, e.g. the concepts of the ancient Greeks, 3000 years ago, who based their thoughts on a small population cannot be applied in 2013 except where populations can be contained into productive working communities with balanced profit and loss centers to ensure sustainability)
Early Life
If we follow the slope from the leftmost start to where it begins to flatten out this can be considered the first period. The first period is characterized by a decreasing failure rate. It is what occurs during the “early life” of a population of units. The weaker units fail leaving a population that is more rigorous.
Useful Life
The next period is the flat bottom portion of the graph. It is called the “useful life” period. Failures occur more in a random sequence during this time. It is difficult to predict which failure mode will occur, but the rate of failures is predictable. Notice the constant slope.
Wearout
The third period begins at the point where the slope begins to increase and extends to the rightmost end of the graph. This is what happens when units become old and begin to fail at an increasing rate. It is called the “wearout” period.
Clone of Clone of BATHTUB MEAN TIME BETWEEN FAILURE (MTBF) RISK
Clone of Bio103 Predator-Prey Model ("Lotka'Volterra")
Tags: Education, Chaos, Ecology, Biology, Population
Thanks 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.
------------------------------------------------------------
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 becomes
dx/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.
PredatorsThe 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.
Clone of MAT375 Clone of Bio103 Predator-Prey Model ("Lotka'Volterra")
Clone of Bio103 Predator-Prey Model ("Lotka'Volterra")
Tags: Education, Chaos, Ecology, Biology, Population
Thanks 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.
------------------------------------------------------------
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 becomes
dx/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.
PredatorsThe 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.
Clone of MAT375 Clone of Bio103 Predator-Prey Model ("Lotka'Volterra")
World4 is a predictive model for world population. Population has grown hyper-exponentially in the last millenium, with the doubling time decreasing from 900 years in 1000 CE to a minimum of ~35 years in 1963 CE. Technology is defined as that which decreases the death rate and/or increases the effective birth rate (i.e. by decreasing infant mortality). Technology grows exponentially, therefore population fits a hyper-exponential (exponent within an exponent). Models for the end of growth are explored using equations that express the ways humans are depleting Earth's biocapacity, the nature of resource depletion, and the relationship between natural resources and human carrying capacity. This simple model, containing just two closed systems, captures the subtle shifts in the population trajectory of the last 50 years. Specifically, hyperexponential growth has given way to subexponential growth. A peak is predicted for the time around 2028. [Bystroff, C. (2021). Footprints to singularity: A global population model explains late 20th century slow-down and predicts peak within ten years. PloS one, 16(5), e0247214.]
Worl4.6 is a clone of World4.5 for exploring changes. WHat happens if there is a delay in the change of the growth rate? What happens if the fertility rate is a function of resource availability? What happens if climate change causes an increased death rate through catastrophes? What happens if policies are enacted to save wildlife?
World4.6
last month
THE 2017 MODEL (BY GUY LAKEMAN) EMPHASIZES THE PEAK IN POLLUTION BEING CREATED BY OVERPOPULATION WITH THE CARRYING CAPACITY OF ARABLE LAND NOW BEING 1.5 TIMES OVER A SUSTAINABLE FUTURE (PASSED IN 1990) AND NOW INCREASING IN LOSS OF HUMAN SUSTAINABILITY DUE TO SEA RISE AND EXTREME GLOBAL WATER RELOCATION IN WEATHER CHANGES IN FLOODS AND DROUGHTS AND EXTENDED TROPICAL AND HORSE LATTITUDE CYCLONE ACTIVITY AROUND HADLEY CELLS
The World3 model is a detailed simulation of human population growth from 1900 into the future. It includes many environmental and demographic factors.
THIS MODEL BY GUY LAKEMAN, FROM METRICS OBTAINED USING A MORE COMPREHENSIVE VENSIM SOFTWARE MODEL, SHOWS CURRENT CONDITIONS CREATED BY THE LATEST WEATHER EXTREMES AND LOSS OF ARABLE LAND BY THE ALBEDO EFECT MELTING THE POLAR CAPS TOGETHER WITH NORTHERN JETSTREAM SHIFT NORTHWARDS, AND A NECESSITY TO ACT BEFORE THERE IS HUGE SUFFERING.BY SETTING THE NEW ECOLOGICAL POLICIES TO 2015 WE CAN SEE THAT SOME POPULATIONS CAN BE SAVED BUT CITIES WILL SUFFER MOST.
CURRENT MARKET SATURATION PLATEAU OF SOLID PRODUCTS AND BEHAVIORAL SINK FACTORS ARE ALSO ADDED
Use the sliders to experiment with the initial amount of non-renewable resources to see how these affect the simulation. Does increasing the amount of non-renewable resources (which could occur through the development of better exploration technologies) improve our future? Also, experiment with the start date of a low birth-rate, environmentally focused policy.
Clone of 2017 Weather & Climate Extreme Loss of Arable Land and Ocean Fertility by Guy Lakeman - The World3+ Model: Forecaster
The relationship between population growth and water consumption of Turkey.
KADİR ÖZBAKIR
OZAN AYDIN
Turkey's population growth and daily water consumption
Von Foerster's doomsday equation says that population growth is hyperbolic because the exponent itself (the growth rate) is a function of population.
Doomsday v.1.0
FORCED GROWTH GROWTH GOES INTO TURBULENT CHAOTIC DESTRUCTION
BEWARE pushing increased growth blows the system!
(governments are trying to push growth on already unstable systems !)
The existing global capitalistic growth paradigm is totally flawed
The chaotic turbulence is the result of the concept and flawed strategy of infinite bigness this has been the destructive influence on all empires and now shown up by Feigenbaum numbers and Dunbar numbers for neural netwoirks
See Guy Lakeman Bubble Theory for more details on keeping systems within finite limited size working capacity containers (villages communities)
BEWARE pushing increased growth blows the system!
(governments are trying to push growth on already unstable systems !)
The existing global capitalistic growth paradigm is totally flawed
The chaotic turbulence is the result of the concept and flawed strategy of infinite bigness this has been the destructive influence on all empires and now shown up by Feigenbaum numbers and Dunbar numbers for neural netwoirks
See Guy Lakeman Bubble Theory for more details on keeping systems within finite limited size working capacity containers (villages communities)
Clone of FORCED GROWTH INTO TURBULENCE
Adapted from Hartmut Bossel's "System Zoo 3 Simulation Models, Economy, Society, Development."
Population model where the population is summarized in four age groups (children, parents, older people, old people). Used as a base population model for dealing with issues such as employment, care for the elderly, pensions dynamics, etc.
Population model where the population is summarized in four age groups (children, parents, older people, old people). Used as a base population model for dealing with issues such as employment, care for the elderly, pensions dynamics, etc.
Clone of Z602 Population with four age groups
Physical meaning of the equations
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 becomes
dx/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.
PredatorsThe 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.
Clone of Prey&Predator
FORCED GROWTH GROWTH GOES INTO TURBULENT CHAOTIC DESTRUCTION
BEWARE pushing increased growth blows the system!
(governments are trying to push growth on already unstable systems !)
The existing global capitalistic growth paradigm is totally flawed
The chaotic turbulence is the result of the concept and flawed strategy of infinite bigness this has been the destructive influence on all empires and now shown up by Feigenbaum numbers and Dunbar numbers for neural netwoirks
See Guy Lakeman Bubble Theory for more details on keeping systems within finite limited size working capacity containers (villages communities)
BEWARE pushing increased growth blows the system!
(governments are trying to push growth on already unstable systems !)
The existing global capitalistic growth paradigm is totally flawed
The chaotic turbulence is the result of the concept and flawed strategy of infinite bigness this has been the destructive influence on all empires and now shown up by Feigenbaum numbers and Dunbar numbers for neural netwoirks
See Guy Lakeman Bubble Theory for more details on keeping systems within finite limited size working capacity containers (villages communities)
Clone of FORCED GROWTH INTO TURBULENCE
Clone of Bio103 Predator-Prey Model ("Lotka'Volterra")
Tags: Education, Chaos, Ecology, Biology, Population
Thanks 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.
------------------------------------------------------------
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 becomes
dx/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.
PredatorsThe 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.
Clone of MAT375 Clone of Bio103 Predator-Prey Model ("Lotka'Volterra")
Clone of Bio103 Predator-Prey Model ("Lotka'Volterra")
Tags: Education, Chaos, Ecology, Biology, Population
Thanks 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.
------------------------------------------------------------
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 becomes
dx/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.
PredatorsThe 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.
Clone of MAT375 Clone of Bio103 Predator-Prey Model ("Lotka'Volterra")
Clone of Bio103 Predator-Prey Model ("Lotka'Volterra")
Tags: Education, Chaos, Ecology, Biology, Population
Thanks 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.
------------------------------------------------------------
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 becomes
dx/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.
PredatorsThe 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.
Clone of MAT375 Clone of Bio103 Predator-Prey Model ("Lotka'Volterra")
This model incorporates several options in examining fisheries dynamics and fisheries employment. The two most important aspects are the choice between I)managing based on setting fixed quota versus setting fixed effort , and ii) using the 'scientific advice' for quota setting versus allowing 'political influence' on quota setting (the assumption here is that you have good estimates of recruitment and stock assessments that form the basis of 'scientific advice' and then 'political influnce' that desires increased quota beyond the scientific advice).
Clone of Fixed Quota versus Fixed Effort
