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Questions tagged [biology]

For questions regarding mathematical concepts with applications to Biology.

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1 answer
424 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 ...
1 vote
0 answers
54 views

Why does a lack of inversion symmetry in 2D pattern formation lead to hexagons?

In reaction diffusion based pattern formation (or other types of pattern formation too, really), it seems that the absence of an inversion symmetry (i.e., if the field $u(x,t)$ is a stable solution/...
0 votes
1 answer
1k views

Model for population growth and finding the equilibrium solutions

A model for population growth is given by: $$\frac{dN}{dt} = f(N) = r N \left( 1-\frac{N}{K} \right) \left( \frac{N}{U}-1 \right) $$ where $r,\ U,$ and $K$ are positive parameters and $U < K$. (a)...
0 votes
0 answers
26 views

Genomic and sum of geometric random variables

In their paper The Maximum of independent Geometric Random Variables as the Time for Genomic Evolutionthe authors noted that if to consider the genomic word of L letters, than the measure of the time ...
0 votes
0 answers
113 views

Family tree in Biology and Category theory

I thought about family tree as a way to understand category theory more easily. The family tree is a diagram that provides detailed information about which family members were born to whom. Like the ...
5 votes
0 answers
756 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 ...
0 votes
0 answers
54 views

Is there a mathematical notation that represents "up until" or "once reached"?

I am a biologist and I need some math help. I am writing a paper about how population density affects life-history traits of a snail. I want to list descriptive statistics of several life-history ...
2 votes
1 answer
505 views

Hopf bifurcation and limit cycles

$$dV/dt=10(V-\frac{V^3}{3}-R+I_{input})$$ $$dR/dt=0.8(-R+1.25V+1.5)$$ Use $I_{input}$ as the relative parameter to prove that there these equations undergo 2 hopf bifurcations and indicate whether ...
0 votes
0 answers
87 views

Solving the PDE $\frac{\partial n}{\partial t} = -v \frac{\partial n}{\partial \alpha} - \mu n $ using a given ansatz

I'm working on exercise 25 of Chapter 10 in Mathematical Models in Biology by Edelstein-Keshet. In the exercise we analyze the following chemotherapy model which accounts for the process of cell aging/...
0 votes
1 answer
52 views

Given a social system in which people are supporters of either a $G$- or a $H$-orientation, calculate the probability to have a $G$-person elected.

In this paper, Majority rule, hierarchical structures, and democratic totalitarianism: A statistical approach, I read: A social system is considered in which people are supporters of either a $G$-...
1 vote
1 answer
53 views

Solving ODE describing negative autoregulation in systems biology

In the paper "Negative Autoregulation Speeds the Response Times of Transcription Networks" (see https://doi.org/10.1016/S0022-2836(02)00994-4) they present an ODE describing negative ...
0 votes
1 answer
147 views

Coalescent theory - Why are coalescent times independent?

I am reading from this book and I want to make sure I understand what is going on. What I get from the book Consider a population of $N$ individuals. The population size ($N$) is constant. select ...
38 votes
2 answers
14k views

The Scutoid, a new shape

The scutoid (Nature, Gizmodo, New Scientist, eurekalert) is a newly defined shape found in epithelial cells. It's a 5-prism with a truncated vertex. The g6 format of the graph is KsP`?_HCoW?T . They ...
3 votes
1 answer
59 views

Modeling the probability of $k$-mer collisions between DNA sequences

Let's imagine that I have a DNA sequence of known origin. Such a sequence can simply be thought of as a string of characters $(A|C|G|T)^l$ where $l$ is the length of the sequence. For purposes of this ...
9 votes
1 answer
638 views

Most Important Challenges in Mathematical Biology

This question is to everyone, but especially to whoever is doing a PhD in Mathematical Biology, to who is teaching Mathematical Biology and finally to who works or researches in this area. What are ...
0 votes
1 answer
67 views

Interpreting replicator dynamic for simplest population model

Suppose the simplest population model where we track the size $y$ of a population: $$\frac{dy}{dt} = ry$$ for a positive constant $r$ and some $y$ such that $y(0) > 0$. For this population model ...
8 votes
5 answers
2k views

Category Theory & Biology

Category theory is becoming more and more used in the following fields (besides others) (1) Quantum physics (e.g. C. Isham and B. Coecke et al) (2) General relativity (e.g. A.K. Guts et al) (3) ...
1 vote
1 answer
82 views

3 species Lotka–Volterra model. Limit cycle

Good day, I have 3 species Lotka–Volterra model. My goal is to determine if there is a limit cycle in the system $$ \left\{ \begin{array}{l} \frac{d c}{d t}=r_c c(1-c)-\frac{c h}{c+\theta_1} \\ \frac{...
0 votes
1 answer
170 views

Recovery Time for Logistic model equation with harvesting

We are given a logistic growth model with constant harvesting as: $\frac{dN}{dt} = rN(1-\frac{N}{K})-Y_0$ We are asked to show that the recovery time for harvesting a yield $Y_0$, $T_R(Y_0)$, ...
0 votes
0 answers
58 views

Can the "escape" trajectory of a gazelle be considered random?

I was watching a video of a gazelle escaping from a cheetah and I wondered: may it's trajectory be considered random? Logically speaking, escaping in a random fashion would make almost impossible for ...
1 vote
0 answers
50 views

persistence of SEIR epidemic model

We have the following stochastic SEIR model $dS=\Lambda - \beta SI - \mu S - \sigma SI dB(t)$ $dE=\beta SI - (\lambda +\mu) E+\sigma SI dB(t)$ $dI=\lambda E-(\gamma +\alpha +\mu) I$ $dR=\gamma I-\mu R$...
1 vote
0 answers
57 views

Showing an endemic steady state is stable

I need to show that the steady state of this non-dimensional model is stable using minimal algebra however I am not sure how to approach this without long lines of working. The model is: $$\frac{dS}{...
6 votes
2 answers
2k 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 ...
1 vote
0 answers
105 views

Finding the eigenvalue/vector of a Leslie Matrix and purpose of eigenvalue/vector

I am currently working on a population dynamics model, in which I have to model the population growth of an animal with the survival rate and fecundity rate. I have set-up a 6 x 6 Matrix below: $$ \...
1 vote
0 answers
46 views

An alternative to the popular Hutchinson population model

As an alternative to the popular Hutchinson population model, which introduces a delay in the per capita growth rate, one can introduce a delay solely in the growth contribution and consider a ...
2 votes
1 answer
126 views

Epidemiology SEI disease model

I have the following model for simple endemic with susceptible (S), exposed (E), and infective (I), $$\frac{dS}{dt}=-\beta SI,$$ $$\frac{dE}{dt}=\beta SI-\delta E,$$ $$\frac{dI}{dt}=\delta E.$$ I have ...
0 votes
0 answers
62 views

Stability Analysis of DFE in terms of R0 Using Ruth-Hurwitz Criterion

I am working with an epidemiological model in the form of a dynamical system of seven ODEs and I am trying to show that the DFE is stable if R0<1. When applying the Routh-Hurwitz criterion, I see ...
2 votes
0 answers
111 views

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 ...
0 votes
0 answers
23 views

Probabilities of a biology experiment

So I was having a biology experiment about Mendel's law, and we are tasked to put 15 pairs of different beads with 4 different colors A, B, C, D (30 x 4 = 120 beads in total) for 2 different ...
0 votes
1 answer
60 views

'Well defined' in a biological context

Question The following model is an approximation of the discrete logistic model $x_{t+1} = f(x_t)$ where $$ f(x) = \begin{cases} \mu x, & 0 \le x \le 1/2,\\ \mu(1-x), & 1/2 \le x \le 1. \...
0 votes
0 answers
18 views

Having two sigmoidal functions with different rate constant, how can I prove that one starts decreasing before the other?

I have two sigmoidal functions that I've fitted to some experimental data. The formula that I'm using to fit the function to my data is the following: $$ f(t) = \frac{A}{1+e^{k(t-t_0)}} $$ Where $k$ ...
0 votes
0 answers
72 views

Probability of a bp sequence in a DNA

Question: DNA is composed of two strands carrying a sequence of nucleotides: adenine (A), cytosine (C), guanine (G) and thymine (T). Nucleotides along the two strands pair (bond) with one another: A ...
2 votes
0 answers
29 views

strategies for looking at the phase space of a system with 6 dimensions

I have a system of odes where the state vector has 6 elements. The system is a population biology model, where I am tracking the evolution of some competing species over time. Now I was trying to ...
5 votes
1 answer
121 views

Confusion About Physical Interpretation of Complex Numbers

Alan Turing's paper, Chemical Basis of Morphogenesis, is about how a symmetrical embryonic stage(like a blastula) can create an asymmetric organism or pattern. He creates differential equations to ...
2 votes
1 answer
81 views

Where to learn about whether a travelling wave solution to the reaction diffusion equation is a pushed or pulled wave?

I'm trying to understand pushed and pulled waves as seen in many biology articles such as: Gene Surfing in Expanding Populations by Hallatschek Spatial gene drives and pushed genetic waves by Tanaka ...
1 vote
1 answer
210 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 ...
1 vote
0 answers
94 views

Can I non-dimensionalize this system of delay differential equations?

I am looking at an application from population dynamics in biology and trying to understand the dynamics of system of delay differential equations below. The model is for a single species with a multi-...
3 votes
2 answers
641 views

SIR infection induced mortality

I am reading the book Modelling Infectious Diseases in Humans and Animals by Matt J Keeling and Pejman Rohani. In it the SIR model is given as, $\frac{dS}{dt} =\mu-\beta{S}I-\mu S$ $\frac{dI}{...
1 vote
1 answer
42 views

How would we measure the non-randomness and compactness of this cubic lattice and corresponding graph?

The genetic code is decrypted into a 4x4x4 cubic lattice shown here: https://www.researchgate.net/publication/361073909_Decryption_and_Topology_of_the_Genetic_Code That cube was used to construct an ...
4 votes
1 answer
146 views

Counting DNA sequences that contain all dinucleotides exactly once

I'm trying to figure out how many unique DNA sequences minimally contain every possible dinucleotide sequence. I'm convinced that this is likely a solved combinatorics question, but I don't have the ...
1 vote
2 answers
64 views

How is this ODE created?

So what I got was this but it doesn't match the answer sheet so not sure what went wrong
0 votes
1 answer
72 views

The French Newbie and the Lotka-Voltera Crazy Idea

I am back to talk with you and to have criticisms on ideas crossing my mind! I am currently working on a disease that is striking french vineyards really hard. I am currently using a model that is ...
1 vote
1 answer
2k views

How to test if data follows a distribution?

Have been given some data and the question says to determine if the data follows any distribution. It says to compare the observed data vs expected graphically and to test further. The distributions ...
3 votes
1 answer
3k views

biology models which uses a system of differential equations

I am trying to find mathematical models used in Biology that uses a system of differential equations. I found the lotka-volterra model and Michaelis-Menten kinetics but I would like to know more than ...
1 vote
1 answer
642 views

Calculate ssgsea score by hand

I hope this is the right place for this question, if not, feel free to suggest more suitable sites. I would like to calculate a single sample gene set enrichment analysis (ssGSEA) score step by step ...
5 votes
2 answers
9k 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 ...
0 votes
1 answer
119 views

Bacteria and logarithms problem. [duplicate]

You are taking a biology class and you are growing a colony of bacteria starting with 5 bacteria. Suppose the colony of bacteria is grow exponentially and can be modeled using the following function: ...
1 vote
0 answers
28 views

Solving chromosome mapping problem using linear algebra?

In biology, we were given a problem in which we had to use "distances" (recombination frequency) between genes in order to create a genetic map of how genes are located relative to each ...
0 votes
0 answers
192 views

Using Jury Conditions to Show Instability

If I want to force an equilibrium point of a discrete dynamical system to be unstable can I just violate one of the conditions for stability stated in the jury conditions $$|\mbox{Trace} (J)| < 1 + ...
2 votes
0 answers
149 views

Converting one form of the Gompertz equation into another

Consider the Gompertz equation that models the dynamics of the population of a single-species $$\frac{dN}{dt}=r_0e^{-\alpha t}N$$ and convert it to the following form $$\frac{dN}{dt}=\alpha N\ln\left(\...

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