Questions tagged [biology]

For questions regarding mathematical concepts with applications to Biology.

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27 views

Deriving logistic growth equation from the exponential

I'm following along with this: https://jdyeakel.github.io/teaching/ecology/section9/ and I just want to make sure my derivation is correct since this website seems to use prime notation not for ...
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1answer
34 views

How to read the surface plots?

I was reading this paper and I could not understand this figure. How do you read these kind of graphs? How to interpret the twists and folds; it's not like heat maps that are intuitive. Any help ...
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1answer
25 views

Logistic decay in mathematical biology

I am studying a course of Mathematical Biology. I am trying to interp the following ODE: $$ \frac{dR}{dt} = \beta R-\gamma(1+\frac{R}{K})R $$ My attempt: $R$ describes a population in time $t$ $\beta ...
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0answers
89 views

Why viruses such as covid-19 are shaped like a sphere from a mathematical point of view [closed]

I can't find anything online about the relationship between spheres and the envelope of a virus from a purely mathematical point of view, what I need to know is why the outer-layer of viruses is a ...
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1answer
27 views

Confused about parameters SIRD model

I'm trying to understand the meaning of the parameters of the following model: \begin{align*} \frac{dS}{dt}&=-\alpha I S+b-dS\\[5pt] \frac{dI}{dt}&=\alpha IS-\delta I-\mu I-dI\\[5pt] \frac{dR}{...
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0answers
172 views

What are the implications of deterministic chaos: useful or detrimental? [closed]

I am new to the concept of chaos theory and as a layman I am struggling to understand what is the significance and implication of chaos in ecological systems such as the chaotic predator prey model. I ...
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0answers
18 views

Dominant eigenvalue of next-generation matrix

I came across an article that says the $R_0$-number is the dominant eigenvalue of the next generation matrix. So I started research and I think I understand what the next generation matrix is. My ...
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0answers
14 views

How to show $p$ that depends on $P$ increases exponentially in the beginning?

Same model as in my last problem, so now if we set $P = 0$ initially, how can we show that $p$ increases exponentially in the beginning? By linear stability analysis? By solving an eigenvalue problem? ...
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0answers
12 views

Under what conditions do either allele become fixed in homozygote dominance random mating model?

I believe this is the correct model I mentioned in the title, where $s, t > 0$: genotype $AA$ $Aa$ $aa$ zygote frequency $p^2$ $2pq$ $q^2$ relative fitness $1+s$ $1$ $1+t$ Now I have to find ...
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1answer
15 views

Find the next gen frequency of diploid selfing and selection model

So we have the genetic diploid model for the spread of an advantageous gene: genotype $AA$ $Aa$ $aa$ zygote frequency $P$ $Q$ $R$ relative fitness $1+s$ $1$ $1$ Now that this model applies to the ...
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0answers
20 views

Determining an ODE with monotonically increasing solution and multiple steady states

I am trying to understand what kind of ODE or system of ODE would result in a monotonically increasing solution with multiple steady states. For lack of better terminology, a solution that increases ...
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0answers
62 views

SIS model with vaccination

For the simple SIS model $$ \begin{aligned} \mathsf{S}′ &= \Lambda − \beta \mathsf{SI} − \mu \mathsf{S} + \gamma \mathsf{I} \\ \mathsf{I}′ &= \beta \mathsf{SI} −(\mu+\gamma)\mathsf{I} \...
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2answers
104 views

SIR model (3 differential equations)

The SIR model for epidemics is Susceptible (not yet infected) -> Infected (have disease and are infectious) -> Recovered (recovered from the disease) and is governed by the system of 3 ...
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0answers
36 views

Check if this system has any attractor

We have the following system: \begin{cases} h'=h(1-h-a_{12}p) & \\ p'=\rho p(1-p-a_{21}h) &, \rho>0, a_{12}>1, a_{21}\in (0,1).\\ \end{cases} We want to see if we have ...
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0answers
34 views

Derivation R0 in a Next-generation Matrix

Following a couple of textbooks for an example, Brauer 2019 Models in Epidemiology, and in whichever paper i read regarding Next-generation-matrices they have defined, R0 as the spectral radius of the ...
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0answers
26 views

Rescale the next system

We have the following system (prey-predator): \begin{cases} h'=rh-\frac{\alpha h}{1+\beta h}p & \\ p'=-ap+\frac{\gamma h}{1+\beta h}p & \end{cases} I have to rewrite it so ...
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1answer
58 views

Travelling Wave Solution For Fisher Equation

I am working with the Fisher equation which I have non-dimensionalised as \begin{equation} \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} +u(1-u) \end{equation} I am looking for ...
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0answers
19 views

What is a time-dependent/non-homogeneous renewal process?

I'm following a lecture in the field of stochastic processes in neuroscience where we are working with objects called time-dependent or non-homogeneous renewal processes. However we never defined ...
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0answers
27 views

Mathematical Biology: Modelling a chemostat

I'm trying to develop differential equations that model a chemostat with two nutrients and one micro-organism. The question states that the volume of the vessel stays constant at amount V throughout ...
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1answer
26 views

The Mean time of Moran Model

I am currently reading an article on Wikipedia on how to derive the mean time before fixation $k_i$ of the Moran Model. But the derivation of this result rests on this equation: $$ k_i^j = \delta_{ij} ...
2
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1answer
35 views

Differential Equations - How to solve a logistic growth model with the added mean

For the general logistic growth model that can be applied to biology and economics $\frac{dP}{dt} = gP(1-\frac{P}{K})$ I know to separate the variables and divide by $P(1-\frac{P}{K})$ resulting in $...
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1answer
24 views

Stability of the modified predator-prey equations?

So I'm facing a modified predator-prey model: $$ \dfrac{dU}{dt} = aU\left(1 - \dfrac{U}{K_U}\right) - cUV,$$ $$ \dfrac{dV}{dt} = ceUV - bV,$$ where $a, b, c, e > 0$. To determine the stability, I ...
3
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1answer
81 views

Coupled Diffusion Equations

I am new to mathematical-biology and I have to solve the following (diffusion-like) equation \begin{align} \frac{\partial a(x,t)}{\partial t}&= D \frac{\partial^2 a(x,t)}{\partial x^2}\\ \frac{\...
4
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0answers
115 views

Simulating a continuous-time Markov branching process

For my MMath dissertation I am exploring the medicinal applications of branching processes, including the Galton-Watson process, Bellman-Harris process ETC. I would like a way for these processes to ...
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0answers
38 views

De Finetti Diagrams Proof

We hav e seen that the genotypic frequencies in a two-allele system can be drawn as points on a de Finetti diagram, sketched above. If the altitude of your equilateral triangle is one, the genotype ...
1
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1answer
53 views

How to calculate fixed points and plot bifurcation diagram for non-linear ODE system

I am trying to understand how to analyse a system of coupled, non-linear ODEs taken from this paper. I want to perform a fixed point analysis and plot a bifurcation diagram to show how fixed points ...
3
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1answer
85 views

Measuring the Shannon Entropy of an ordered sequence

I have 927 unique sequences of the numbers 1, 2 and 3, all of which sum to 12 and represent every possible one-octave scale on the piano, with the numbers representing the intervals between notes in ...
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0answers
21 views

Physical meaning of persistence and extinction definition

I came to know that the mathematical definitions for weak persistence and extinction of species are $$\limsup_{t \to \infty} \frac{1}{t} \int_{0}^{t} x(s) ds >0 ~~\mbox{and}~~ \limsup_{t \to \infty}...
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1answer
30 views

Using flux to calculate concentration in a given region

I am told that the diffusion equation for radially symmetric flow is $\frac{∂u}{∂t}=\frac{1}{r}\frac{∂}{∂r}\left(rD\frac{∂u}{∂r}\right)$. This is with flux $J=-D\frac{∂u}{∂r}$. I can show that the ...
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0answers
24 views

Extinction time distribution in stochastic system

I have seen in different work of stochastic predator-prey model that extinction time follows a lognormal distribution and I have also tested it in a model. Can anyone please tell me what is the reason ...
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0answers
77 views

Mathematical Biology Question on Two Systems.

Species x(t) and y(t) population models are as follows: ẋ = βx(1 - x/κ) - αxy ẏ = γxy - δy (Assuming α,β,γ,κ > 0) 1.) How do the populations behave in the absence of each other? 2.) What type of ...
1
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1answer
57 views

The Volterra-Lotka model (predator prey equations) — Linearization ODE

Q: Find all fixed points of the equation, linearize the equation, substitute the origin point $(0, 0)$ into it and solve the linear version of Volterra-Lotka model. The system looks like this (where $...
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0answers
29 views

How to approach differential equations with separate x and y?

I am studying population modelling, and most of the studies I looked at was in this format, which I was not familiar with. Previously I was looking at basics such as logistic models, delayed logistic ...
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0answers
10 views

Summary statistics of dispersal kernels.

The exponential dispersal kernel is defined as $\frac{m^2}{2\pi}e^{-mr}$, what do the values of m and r represent? We can also define the effective area and mean dispersal distance as $\frac{8\pi}{m^2}...
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1answer
49 views

Equilibrium of simple SIS model

I have a simple SIS model with a non-constant population size due to disease-related mortality: \begin{aligned} \frac{dS}{dt} &= a I - b S I\\\\ \frac{dI}{dt} &= b S I - a I - c I \end{...
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0answers
29 views

Weighting conditions with different number of measurements in parameter fitting

I am currently thinking of the following problem: I have measured time series of a (biological) system under 2 different experimental conditions. Both conditions are equally important. In condition c1 ...
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0answers
45 views

Using ODE to model a discrete phenomenon

SETTING In this paper the authors study the effects of tourism in a national park in Austria on a endangered species of bird. They model the species size (number of breeding couples) with a continuous ...
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0answers
8 views

Example of informative and independent censoring

Does anyone of you have practical example of censoring, that is both independent and informative? For example the time to event may have $Exp(3 * \lambda)$ distribution and censoring have $Exp(\lambda)...
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0answers
94 views

Which mathematics areas are important to be strong in for research in mathematical modelling of neuroscience/human psychology?

I’m guessing ODEs/PDEs would play a big role - I wonder exactly what methods from ODE/PDEs are used in this kind of research? I also wonder whether stochastic processes and stochastic differential ...
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1answer
89 views

How many milligrams of the medication remain in bloodstream after $8$ hours?

The rate at which a particular medication leaves an individual's bloodstream is proportional to the amount of this medication that is in the bloodstream. An individual takes $275$ milligrams of the ...
1
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1answer
54 views

SIRS Model Analysis

Consider the SIRS model given by \begin{align} \frac{dS}{dt}&=-aSI+cR, \\ \frac{dI}{dt}&=aSI-bI, \\ \frac{dR}{dt}&=bI-cR. \\ \end{align} Here $a,b,$ and $c$ are constants and $N=S+I+R$ ...
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2answers
847 views

What are differences between Geometric, Logarithmic and Exponential Growth?

At past I have read in some ecology text that geometrical, logarithmic and exponential growths are not exactly the same thing; and there were various equations for them. (The book is not available to ...
3
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1answer
98 views

How to approximate the peak of an epidemic (depending on epidemiological parameters)?

[I posted a follow-up question at MathOverflow.] Numerical solutions of the SEIR equations (describing the spreading of an epidemic disease) $\dot{S} = - N$ $\dot{E} = + N - E/\lambda$ $\dot{I} = +...
1
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1answer
31 views

Understanding the $\gamma$ rate of the SIR model

Relating the SIER model, I am trying to understand the intuition of the $\gamma$ paramater. This paramater is the recovery rate. $\gamma$ is fixed and biologically determined. Some authors conider for ...
2
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1answer
53 views

An unexpected pair of almost-Fibonacci and Tribonacci series

I use a modification of the SEIR model of epidemic spread which yields - for me totally out of the blue - for special parameters an astoninglishly good approximation of the Fibonacci series with a ...
4
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1answer
205 views

Fibonacci and the spreading of viruses

I tried to understand better the spreading of a virus on a case-by-case basis, not by differential or difference equations as in SIR and SEIR models with possibly non-integer rates and time constants. ...
0
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0answers
21 views

What would be the steady state of x in following system X -k-> 2X, 2X -k*v(t)-> ∅ when v(t) = vo (const)?

I have a system of a mass action type. $\require{AMScd}$ \begin{CD} X @>{\text{$k$}}>> 2X, 2X @>{\text{$k\cdot v(t)$}}>> ∅ \end{CD} When $v(t) = v_0(const)$, what is the stable ...
0
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1answer
104 views

How to define complexity?

Before Shannon's 1948 paper, most people hadn't realized that entropy, information, and their thermodynamics are closely related. I believe the concept of complexity today is also reducible between ...
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0answers
9 views

Given an alignment, tree and mutation parameter, factorize the likelihood P(D|g, µ)

In bioinformatics, for a given alignment D, tree g and mutation parameters θ, how do we find the likelihood P(D|g, µ) to make it calculation tractable? Several assumptions of independence have to be ...
1
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1answer
61 views

Characteristic method for McKendrick-von Foerster age structure model

The well-known age-structure/transport equation by McKendrick and von Foerster has the form: $$\frac{\partial }{\partial t} u(t,x) + \frac{\partial }{\partial x} u(t,x) = -\lambda u(t,x), \quad \quad \...

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