Questions tagged [biology]
For questions regarding mathematical concepts with applications to Biology.
401
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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/...
- 1,087
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1
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50
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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 $G$-...
- 7,727
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1
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48
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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 ...
- 13
3
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1
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52
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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 ...
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Adjusting death rate based on known half-life
Most health applications use exponential decay for estimating the death rate. This assumes at every age, there is an equal chance of dying. Is there a simple way to slightly improve this assumption by ...
- 1,797
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51
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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 ...
- 191
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16
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How can I replicate the model of a paper that describes an animal's foraging radius?
I'm trying to follow the algebra of a paper that describes a model used to estimate a bird's foraging radius. It takes an energetics approach.
$T$ is the foraging cycle, $E_{ma}$ is adult basal ...
- 223
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47
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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
$$
\left\{
\begin{array}{l}
\frac{d c}{d t}=r_c c(1-c)-\frac{c h}{c+\theta_1} \\
\frac{...
- 11
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votes
1
answer
108
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Recovery Time for Logistic model equation with harvesting
We are given a logistic growth model with constant harvesting as:
$\frac{dN}{dt} = rN(1-\frac{N}{K})-Y_0$
We are asked to show that the recovery time for harvesting a yield $Y_0$, $T_R(Y_0)$, ...
1
vote
0
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42
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persistence of SEIR epidemic model
We have the following stochastic SEIR model
$dS=\Lambda - \beta SI - \mu S - \sigma SI dB(t)$
$dE=\beta SI - (\lambda +\mu) E+\sigma SI dB(t)$
$dI=\lambda E-(\gamma +\alpha +\mu) I$
$dR=\gamma I-\mu R$...
- 15
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51
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Showing an endemic steady state is stable
I need to show that the steady state of this non-dimensional model is stable using minimal algebra however I am not sure how to approach this without long lines of working.
The model is:
$$\frac{dS}{...
1
vote
0
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90
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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:
$$
\...
- 11
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0
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41
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An alternative to the popular Hutchinson population model
As an alternative to the popular Hutchinson population model, which introduces a delay in the per capita growth rate, one can introduce a delay solely in the growth contribution and consider a ...
- 8,910
2
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1
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101
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Epidemiology SEI disease model
I have the following model for simple endemic with susceptible (S), exposed (E), and infective (I),
$$\frac{dS}{dt}=-\beta SI,$$ $$\frac{dE}{dt}=\beta SI-\delta E,$$ $$\frac{dI}{dt}=\delta E.$$
I have ...
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50
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Stability Analysis of DFE in terms of R0 Using Ruth-Hurwitz Criterion
I am working with an epidemiological model in the form of a dynamical system of seven ODEs and I am trying to show that the DFE is stable if R0<1. When applying the Routh-Hurwitz criterion, I see ...
- 1
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How to calculate a new pathogen start time the outbreak in humans by a phylogenetic tree with math method？
Now we have a phylogenetic tree built from the virus sequences from human patients (red) and
infected chickens (blue).
We may calculate the mutuation rate via the phylogenetic tree.
But the problem is,...
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106
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Biological meaning of "eigenvalue" in DDE-based population model
For example, consider a Verhulst model with delay
$$ \boxed{\dot N(t) = r N(t) \left( 1 - \frac{N(t-T)}{K} \right)} $$
where $r$ gives the reproduction rate and $K$ means the carrying capacity of the ...
- 21
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0
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23
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Probabilities of a biology experiment
So I was having a biology experiment about Mendel's law, and we are tasked to put 15 pairs of different beads with 4 different colors A, B, C, D (30 x 4 = 120 beads in total) for 2 different ...
- 315
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1
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60
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'Well defined' in a biological context
Question
The following model is an approximation of the discrete logistic model $x_{t+1} = f(x_t)$ where
$$
f(x) = \begin{cases}
\mu x, & 0 \le x \le 1/2,\\
\mu(1-x), & 1/2 \le x \le 1.
\...
- 305
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16
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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
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answers
46
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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
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29
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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 ...
- 2,159
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1
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119
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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
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1
answer
70
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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 ...
- 41
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76
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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-...
- 2,159
1
vote
1
answer
39
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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 ...
- 111
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1
answer
118
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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 ...
- 113
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2
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64
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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
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1
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70
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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 ...
- 11
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votes
1
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102
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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:
...
- 79
1
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28
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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 ...
- 11
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0
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155
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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
$$|\mbox{Trace} (J)| < 1 + ...
- 1
2
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0
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132
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Converting one form of the Gompertz equation into another
Consider the Gompertz equation that models the dynamics of the population of a single-species $$\frac{dN}{dt}=r_0e^{-\alpha t}N$$ and convert it to the following form $$\frac{dN}{dt}=\alpha N\ln\left(\...
- 375
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197
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Comparing solutions of logistic equation and Gompertz equation
Considering the Gompertz equation, $\frac{dN}{dt}=r_oe^{-\alpha t}N$, and the logistic growth equation, $\frac{dN}{dt}=rN(1-\frac{N}{k})$. How is the carrying capacity from Gompertz equation differ ...
- 375
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vote
1
answer
244
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Question in population dynamics using exponential growth rate equation
Given population doubles in 20 minutes, what is intrinsic growth rate r?
Attempt: Given population doubles, using exponential growth rate we have $\frac{dN}{dt}=2N$ so $N(t)=N_0e^{2t}$ therefore r=2, ...
- 375
4
votes
2
answers
124
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Comparison of bird diet at two different nests
I have a list with the different prey types and quantities that birds at Nest $1$ gave to their fledglings. I also have the same information for another nest of the same bird species.
Suppose I have
<...
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0
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49
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On the initial conditions for bacterial population growth
I want to prepare an example of a differential equation for bacetrial growth.
We have $\frac{dy}{dt}=ry(t)$
Multiplying by dt and moving y(t) over to the other side we get $\frac{1}{y(t)}dy=rdt$, ...
- 2,574
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1
answer
77
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How do you call this system?
Is there a specific name for a dynamical system that depends on the relative indexation $i\pm k$ for some $k$? For example, consider the following dynamical system defined on a ring of cells by
$$
\...
- 3,095
3
votes
1
answer
68
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Turing's Symmetries
In Alan Turing's classic paper on the Chemical Basis of Morphogenesis, the following bilateral and left-right symmetries are defined as follows:
Might be an misunderstanding of English from my side, ...
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Can I use the number of particles rather than concentration in biological differential equations?
(This question is not specific to biology, but I was just wondering about a common way that people view differential equations in biology.)
Some simple biological differential equations take the form $...
0
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1
answer
72
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Finding units for constants in ODE's
$$ \begin{array}{r c l} \frac{dN}{dt} & = & rN\left(1-\frac{N}{K}\right)-\alpha \frac{NP}{\beta P + \gamma N} \\
\frac{dP}{dt} & = & \epsilon \frac{NP}{\beta P + \gamma N} - \delta P \...
0
votes
1
answer
101
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How to solve simple differential equation (biology)
First of all, I am a biologist and I am not really knowledgeable in mathematics. Thus, I apologize if what I am asking is naive or not fully explained.
I am trying to solve analytically a differential ...
-1
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1
answer
130
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Predator-Prey Model [closed]
Can anyone explain the biological interpretation on the right hand side of these equations please?
$$ \begin{array}{r c l} \frac{dN}{dt} & = & rN\left(1-\frac{N}{K}\right)-\alpha \frac{NP}{\...
0
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0
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24
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What is Iteration Number in modeling?
I am a biologist. Referring to mathematical modeling for biological data in the literature.
Could someone explain what is the "Iteration Number" of modeling and the meaning of the up and ...
- 175
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30
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Non-trivial, time-independent solution at long times for a separable solution of an age-structured model
I'm taking an introductory course on mathematical modelling in biology and am completely at a loss with this question. We are considering the PDE for a population of cells $n$ with age $a$ over time $...
- 358
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168
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Simplify the following differential equation by variable scaling
We have: $\frac{dx}{dt}=\frac{a}{x+b}, a>0,b>0$.
We make substitutes: $x\to\alpha h, t\to\beta \tau$.
Then the differential equation will be:
$\frac{\alpha}{\beta}\frac{dx}{dt}=\frac{a}{x+b}$, ...
- 172
1
vote
1
answer
39
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Probability of having an all-heterogeneous genotype
Find the probability that the genotype of an offspring will be $\mathrm{AaBbCcDdEeFfGgHhIiJj}$ from the cross $\mathrm{AaBbCcDdEeFfGgHhIiJj \times AaBbCcDdEeFfGgHhIiJj}$.
Making a Punnett square for ...
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votes
1
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147
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Can SIR models explain reduction in COVID case numbers in Florida and Texas, despite lack of restrictions?
This is a question about SIR or SEIR models and understanding the trajectory of the covid infections in Florida and Texas in September 2021. This is a genuinely academic question, and not meant to be ...
- 2,159
1
vote
0
answers
27
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Expected Value for distinct substrings of length $k$
Genomes consist of 4 nucleobases (A, C, G, T). If we take genome substrings of length $k$ (called $k$-mers), we have $4^k$ theoretically possible substrings. If we have a genome of length $l$ (meaning ...
2
votes
1
answer
133
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Equivalence of Leaky Integrator and Low-Pass Filter Decision Models
I'm working with a decision making model from the cognitive neuroscience literature (the Urgency Gating Model) and have found two different implementations. My concern is that these two ...
- 131