Questions tagged [population-dynamics]
For questions related to mathematical models to study the size and age composition of populations as dynamical systems.
42
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Mapping the Lotka-Volterra System to the Replicator System
To Show: if all $r_i$ are equal in the n-dimentional Lotka-Volterra equation then the $x_i=y_i(y_1+...+y_n)^{-1}$ satisfies the replicator equation.
This question is equivalent to Exercise 7.5.2 of ...
- 318
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0
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54
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Population growth model
I tried to create very simple population model with only two variables; population and food.
I want food to be a function of population, every person is working on making food and creating some small ...
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21
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Logistic growth decreasing
In a city, the initial population is
$$P(0)=70281$$
and 10 years after,
$$P(10)=66277$$
I have to find $P(t)$ using the logistic model with max value equal to 80000, which gives the following equation
...
- 2,084
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Stability of normal state in chemostat model
The chemostat model proposed by monod was given by,
$$
\begin{align}
\frac{dx}{dt}&=[K(c)-D]x\\
\frac{dc}{dt}&=D[c_0-c]-\frac1yK(c)x
\end{align}
$$
where $x(t)$ is the population of micro-...
- 701
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25
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Basic reproduction number for complicated diseases model
For an epidemic model, the basic reproduction number is defined as the average number of new infections (e.g. infectious individuals) generated by one infectious individual in an otherwise completely ...
- 701
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1
answer
70
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Find Steady state, determine stability and solve a negative feedback system
Question: Use geometric arguments for the model of a negative feedback system,$$\frac{dx}{dt}=\frac{A\theta^2}{\theta^2+x^2}-\gamma x$$where $A, \theta$ and $\gamma$ are positive constants.
$(a)$ How ...
- 701
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1
answer
150
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Deriving the derivative of The Gompertz trajectory function
Can someone show me the steps involved in computing the derivative of the Gompertz trajectory function for $N(t)$?
$$ N(t) = K \cdot \exp\Big( e^{-\theta \cdot t} \cdot \ln \big(\frac{N_0}{K}\big)\Big)...
- 23
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1
answer
144
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Finding limit from a differential equation
So to define a new population growth model, this is the (Cauchy problem for a) differential equation of growth I am considering:
$$
\begin{cases}
\dfrac{\mathrm{d}N}{\mathrm{d}t} = rN(t) - r\dfrac{N^2(...
1
vote
0
answers
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What is the equation and area under curve for Covid load dynamics?
Covid virions on infection, replicate exponentially and once the body's defense system starts attacking it then it also seems to decrease exponentially.
Source
The time period when the PCR test is ...
- 111
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vote
1
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Modelling exponential growth with individual limited lifetime/death
The Wikipedia article on Diatoms states that:
an assemblage of living diatoms doubles approximately every 24 hours by asexual multiple fission; the maximum life span of individual cells is about six ...
- 150
2
votes
1
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How to mathematically model my population growth simulation
In high-school we learn to model population growth as an exponential, but we know that this is different from reality because population growth seems to hit as asymptote as some point due to limited ...
- 769
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Intrinsic Growth Rate of Beverton-Holt Model
I am reading ahead in some lecture notes that give the Beverton-Holt model with zero harvest:
$r_t=\dfrac{ar_{t-1}}{b+r_{t-1}}$.
It states that under a standard assumption of $a>b>0$, the '...
- 115
3
votes
1
answer
96
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Method of characteristic to solve Sharpe-Lokta model
The conservation law for the population is,
$$
\underbrace{\frac{\partial}{\partial t} x(t,a) + \frac{\partial}{\partial a} x(t,a)}_{\text{directional derivative}} = -\mu(a) x(t,a) dt\tag1
$$
where $x(...
- 701
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votes
2
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Extinction of non-dominant species in generalized competitive Lotka-Volterra systems
I am studying the generalized $n$-species competitive Lotka-Volterra system where populations of species $i$ are defined by the standard differential equation:
$$ \dot x_i = f_i(\mathbf{x}) := x_i \...
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Spectral graph theory for population projection matrices
Consider a population structured into $s$ categories, and a matrix $\mathbf{M}$ of size $s\times s$, that projects deterministically the population vector $\mathbf{n}$ of length $s$.
All elements of $\...
- 147