Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [population-dynamics]

For questions related to mathematical models to study the size and age composition of populations as dynamical systems.

Population dynamics An excerpt from Wikipedia: Population dynamics has traditionally been the dominant branch of mathematical biology, which has a history of more than 220 years, although over the last century the scope of mathematical biology has greatly expanded.

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How to calculate compound interest with shrinking interest rates?

The question arised when I was thinking about population growth. If a population grows by 1% we can simply plug into to the formula: $$ \text{final population}= P \cdot (1+p)^n $$ where $P$ is the initial population, $p=0.01$ in the example and $n$…
timtam
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Find condition for Co-existence in two species Competition model

If $b_1b_2-c_1c_2=0$ then find the condition for Co-existence in two species Competition model The two-species competition model is, $$ \begin{align} \frac{du}{dt}&=(a_1-b_1u-c_1v)u\\ \frac{dv}{dt}&=(a_2-b_2v-c_2u)v \end{align} $$ One equilibrium…
WhyMeasureTheory
  • 895
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Find steady state of AIDS epidemic model

AIDS epidemic in a homosexual population The following diagram shows the AIDS epidemic in a homosexual population: Then the model can be described by $$ \begin{gathered} d X / d t=B-\mu X-\lambda c X \\ d Y / d t=\lambda c X-(v+\mu) Y \\ d A / d…
WhyMeasureTheory
  • 895
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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 $x(t,a)$ is the density of individuals of age $a$…
N00BMaster
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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: $$ \dot x_i = f_i(\mathbf{x}) := x_i \left( 1 - \sum_j a_{ij}x_j \right) $$ where all…
Johannes Nauta
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Help Finding a Traveling Wave Solution

I am looking for traveling wave solutions of \begin{align} \frac{\partial U}{\partial t} &= AU\left(1-\frac{U}{K}\right)-BUV+D_{1}\nabla^{2}U \\ \frac{\partial V}{\partial t} &= CUV-DV+D_{2}\nabla^{2}V \end{align} Where $A,B,C,D,K,D_{1}$,and $D_{2}$…
Iesha Patterson
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Proof for positivity of solutions of an ODE system

I have an ODE system that takes a mathematical model describing the dynamics between HCV and the immune system. My question is about the proof that the solution of the ODE system is positive if the initial conditions are all positive. I tried this…
Anas TABET
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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 surplus. However, food production is limited by…
klobaska soslaninou
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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 resources. I wanted to see if I could recreate…
Dargscisyhp
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What is the equivalent formulation for the Logistic Growth model in a population dynamics/branching processes setting?

I am reading a recently released book entitled Branching Processes: Variation, Growth, and Extinction of Populations by Haccou, Jagers, and Vatutin. One key concern of the book is the connection between population models and discrete dynamical…
krishnab
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Biological meaning of "eigenvalue" in DDE-based population model

For example, consider a Verhulst model with delay $$ \boxed{\dot N(t) = r N(t) \left( 1 - \frac{N(t-T)}{K} \right)} $$ where $r$ gives the reproduction rate and $K$ means the carrying capacity of the environment. The non-trival equilibrium is…
m1ssing
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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 positive is 14 days, let it be $t$. In the above…
Hari Kumar
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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 days So a simple exponential equation doesn't…
2080
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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 $\mathbf{M}$ and of $\mathbf{n}$ are real and…
Christophe
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Properties of the bifurcation diagram for the logistic function

Once the bifurcation diagram has been plotted ($x_{n+1}=rx_n(1-x_n)$), there are 3 elements or properties that I don't know haw to explain, and I have not found any article where they are explored. Those are: Islands of stability; or the spaced…
Minerva González García
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