Part II Mathematical Biology - Lent 2017
Part II Mathematical Biology - Lent 2017Copyright c 2019 University of Cambridge. Not to be quoted or reproduced without permissionLecturer: Prof. Julia Gog (jrg20@cam.ac.uk)(this version September 2019 - some typos corrected)Contents0 Introduction31 Deterministic systems, no spatial structure51.1 Single population models . . . . . . . . . . . . . . . . . . . . . . . . . .51.1.1 Simple birth and death models . . . . . . . . . . . . . . . . . . .51.1.2 Delay models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .81.1.3 Populations with age structure . . . . . . . . . . . . . . . . . . .191.2 Discrete time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .251.2.0 Revision: 1-D stability in difference equations (maps). . . . . .251.2.1 The logistic map . . . . . . . . . . . . . . . . . . . . . . . . . . .261.2.2 Higher order discrete systems . . . . . . . . . . . . . . . . . . .301.3 Multi-species models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .351.3.0 Revision: 2-D stability in continuous time . . . . . . . . . . . . .351.3.1 Competition models . . . . . . . . . . . . . . . . . . . . . . . . .361.3.2 Predator-prey models . . . . . . . . . . . . . . . . . . . . . . . .401.3.3 Chemical kinetic models . . . . . . . . . . . . . . . . . . . . . . .461.3.4 Epidemic models . . . . . . . . . . . . . . . . . . . . . . . . . . .501.3.5 Excitable systems . . . . . . . . . . . . . . . . . . . . . . . . . .591
Copyright c 2019 University of Cambridge. Not to be quoted or reproduced without permission2 Stochastic systems642.0 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .642.0.0 Revision: discrete probabilities and generating functions . . . . .642.0.1 Why bother? . . . . . . . . . . . . . . . . . . . . . . . . . . . . .652.0.2 The first step . . . . . . . . . . . . . . . . . . . . . . . . . . . . .662.1 Discrete population sizes . . . . . . . . . . . . . . . . . . . . . . . . . .682.1.1 Single populations . . . . . . . . . . . . . . . . . . . . . . . . . .682.1.2 Extinction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .772.1.3 Multiple populations . . . . . . . . . . . . . . . . . . . . . . . . .802.2 Continuous population sizes . . . . . . . . . . . . . . . . . . . . . . . . .842.2.1 Fokker-Planck for a single variable . . . . . . . . . . . . . . . . .842.2.2 Multivariate Fokker-Planck . . . . . . . . . . . . . . . . . . . . .883 Systems with spatial structure973.0 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .973.1 Diffusion and growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . .993.1.1 Linear diffusion in finite domain . . . . . . . . . . . . . . . . . . .993.1.2 Linear diffusion in infinite domain . . . . . . . . . . . . . . . . . . 1033.1.3 Nonlinear diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 1123.2 Travelling waves in reaction-diffusion systems . . . . . . . . . . . . . . . 1163.2.0 General F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1163.2.1 Fisher’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1173.2.2 Bistable systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1263.3 Spatial instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1303.3.1 Chemotaxis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1303.3.2 Turing instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . 1352
0IntroductionCopyright c 2019 University of Cambridge. Not to be quoted or reproduced without permissionAcknowledgementsMuch of the content and the starting point for these notes comes from Prof. PeterHaynes’s version of the notes from 2012. Any typos or errors are, however, likely to bethe fault of this author.PracticalitiesThis are intended as archived version of the notes from the Part II Mathematical Biologycourse, as lectured by me (Julia Gog) in Lent 2017, within Part II of the MathematicalTripos, University of Cambridge. For Part II students in future years, these notes mightbe useful as an extra resource perhaps for reading ahead over the summer, but pleasedo not use them as a replacement for attending lectures or working from the presentlecturer’s resources. Lectures are good, you should go to them: you’ll always learn abit more, see something in a different way or build more intuition for what is really goingon.There are exercises in green boxes throughout these notes. These were set in lecturesand intended as being additional to the usual examples sheets. These exercises shouldbe fairly doable without any help from a supervisor, and I recommend doing them asyou work through the notes. There exist a full set of ‘solutions’: you might find thesewherever you found these notes: these can be used to check you are working alongthe right lines, or to see what was intended.For anyone who uses the notes, I hope these notes are interesting and useful. Mostof all, I hope they inspire you to explore mathematical biology further. If you have anycomments or corrections, please do email me: jrg20@cam.ac.uk. I might not respondimmediately (or at all, sorry!), but feedback will be useful in updating and amendingthese for future use.Preparation for this coursePart II Dynamical Systems is ‘helpful’ for parts of this course but certainly not essential. If you did not do Dynamical Systems, then it might be wise to do a little revisionof parts of Ia Differential Equations: stability of equilibria of discrete and continuoustime systems (Jacobians, saddles/focus/node, phase-plane diagrams). Indeed ‘Ordinary Differential Equations’ by Robinson (see schedules for Ia Differential Equations)chapters 32 and 33 (‘coupled nonlinear equations’ and ‘ecological models’) will put youright on track for this course. The middle part of the course on stochastic systems willuse some knowledge from Ia Probability, including generating functions. It would be a3
good idea to revise separable solutions from Ib Methods for the last part of the courseon diffusion.Interesting readingCopyright c 2019 University of Cambridge. Not to be quoted or reproduced without permissionNone of these are essential to follow this course, but should be of interest: J.D. Murray Mathematical Biology (3rd edition) (see schedules) - the classic texton mathematical biology, covering a range of applications D. Neal Introduction to Population Biology - much overlap with this course inmathematical detail, but explores the biological principles in rather more depthand includes many real examples. Should be completely readable by you duringor after this course. Mathematics is biology’s next microscope, only better; Biology is mathematics’next physics, only better - article by Joel E. Cohen in PLoS Biology 2004 DOI:10.1371/journal.pbio.00204394
1Deterministic systems, no spatial structure1.1Single population modelsCopyright c 2019 University of Cambridge. Not to be quoted or reproduced without permission1.1.1Simple birth and death modelsThe simplest model?Let x(t) be population size as a function of time t. Assume that the number of offspringproduced per individual per unit time is a constant b 0. Similarly assume that thedeath rate (number of deaths per unit time per individual) is a constant d 0.x(t δt) x(t) b x δt d x δtDivide by δt and take the limit as δt 0.dx (b d)x rxdtwhere r b d.Solution is x(t) x0 ert , where x(0) x0 , so the population grows indefinitely if r 0and decays towards zero (implying extinction) if r 0.Exercise 1: In the case when r 0, find the half-life of the populationExercise 2: Actually, this simple model is pretty good for invasions of new populations. Suppose a new disease is discovered and there are 1000 cases lastweek and 1500 cases this week, roughly when did the disease first appear?Note that in a deterministic system, only the difference between b and d matters,e.g. b 21, d 20 gives entirely the same dynamics as b 1, 000, 001, d 1, 000, 000.These will differ in an analogous stochastic model (the ones with higher rates will fluctuate wildly).Birth and death rates depend on population sizeRather than constant, allowthe number of offspring per individual per unit time to depend on population size, a(x),and similarly the death rate b(x). Then we have:dx [b(x) d(x)]xdt5
Again, only the difference between birth and death rates matter in the deterministicsystem.Copyright c 2019 University of Cambridge. Not to be quoted or reproduced without permissionTypically, one might expect the birth rate per capita to decrease and/or death rate toincrease for very large population size, as resources become scarce. For examplehere we could take the birth rate to be constant (b(x) B) and the death rate to beproportional to population size (d(x) D x):dx [B D x]xdtBy rescaling the population size and renaming parameters, we have the logistic equation:dx αx(1 x)dtExercise 3: Find the rescaling.For x 1, births outnumber the deaths and the population grows. For x 1, theopposite occurs and the population shrinks. The equilibrium population size (scaled) isone.The logistic modeldx αx(1 x)dtThis is easy to solve:Z1dx x(1 x)Z11x dx log C αtx 1 x1 xSo putting x x0 at t 0 we have:xx0 αt e1 x1 x0Which rearranges to:x x0 eαt(1 x0 ) x0 eαt6
Copyright c 2019 University of Cambridge. Not to be quoted or reproduced without permissionReassuringly, our steady population size is there: x0 1 gives x(t) 1. Also, it isalways sensible to check zero initial conditions: x0 0 gives x(t) 0.Exercise 4: show that the solution to the logistic equation can be rewritten forsome t0 as: 21 12 tanh 12 α(t t0 )for x0 1x 1 1 coth 1 α(t t ) for x0 10222Note that for positive initial population size (x0 0), x 1 as t (from above ifx0 1 and from below if x0 1). There is a stable equilibrium, achieved for all positiveinitial conditions. The zero equilibrium x 0, b is unstable.One-dimensional stability recapConsider:dx f (x)dtThe steady-states are the values of x for which f (x ) 0. These may be interchangeably referred to as equilibria, fixed points, steady states or constant solutions. Stabilityis determined by behaviour near the fixed point, which can be found by linearisationaround x . Set x(t) x (t). Then:dxd f (x ) f (x ) f 0 (x ) O( 2 ) {z } {z }dtdt 0ignoreHence:d ' f 0 (x ) which has solutiondt (t) ' 0 exp[f 0 (x )t].So , the perturbation away from x grows if f 0 (x ) 0 (unstable) and shrinks if f 0 (x ) 0 (stable).In practice, just check sign of f 0 at fixed points. For simple biological models, this canusually be done easily by plotting f .Exercise 5: check stability of the fixed points of the logistic model7
Copyright c 2019 University of Cambridge. Not to be quoted or reproduced without permission1.1.2Delay modelsSo far, we have x0 (t) depending on x(t), i.e. the instantaneous current population size.Of course this is not always realistic. For example, offspring are not really producedinstantaneously, there may be a significant gestation period, or time for eggs to hatch.Even then, new offspring may need further time to mature to adulthood, before they canin turn produce offspring. So, we might want x(t) to denote adults, and births and/ordeaths may depend on the population size at some past time point. In physiologicalmodels, there is often some form of delay, for example heart rate does not respondinstantly to exercise. In biochemical signalling, there can be many steps between atrigger and effect, which can sometimes be modelled relatively simply as a time lag.end oflecture 1Mathematically, this leads to delay-differential equations (DDEs). Here is an example,the Hutchinson-Wright equation, which can be viewed as an extension to the logisticequation:dx α x(t) [1 x(t T )]dtwhere the delay time T is a new parameter in the model (assume T 0, note T 0was logisitic equation).We can analyse its dynamics with much the same ideas as before: find the interestingfixed points and look at their stability by considering a small perturbation. Clearly x(t) 1 is still the non-trivial steady state. Now set x(t) 1 (t) and sub in:d α(1 (t))( (t T ))dtd α (t T ) O( 2 )dtAnd drop O( 2 ) from here.This is still linear, so reasonable to seek a solution of the form 0 est :s α e sT(1)We would like know the solutions for s. We see that if T 0, then this just returnss α, which corresponds to the stable fixed point of the logistic equation. If T 0,then we need to look a bit more carefully.First we might reasonably seek real s solutions. Rearranging:sT esT αT8
Consider the shape of the LHS as a function of sT . It has a single minimum at sT 1when the LHS is equal to e 1 . So, there are negative real roots for αT e 1 and noreal roots otherwise.Copyright c 2019 University of Cambridge. Not to be quoted or reproduced without permission ( ) -1-e-1If we look at the solution near sT 0, for small αT , the gradient is approximately 1, sowe have sT αT so this is a continuation of the solution s α, which is what wewould have got with the logistic equation.So far we have only considered real roots for s, but we might (correctly) suspect therecould be complex roots of 1 for s. What would this mean? Our perturbation wouldfollow 0 est , so a complex solution would just give a solution that grows or decays butwith oscillations (think back to complimentary functions in second order linear ODEs).We are now interested in the sign of the real part of s. If Re(s) 0 we say it is unstable,if Re(s) 0 we say it is stable. It is not usually possible to solve explicitly for s, but wecan see now that it would be sensible to find when stability might change, i.e. Re(s) 0.Now lets seek complex roots of (1) by setting s σ iω (where σ and ω are the realand imaginary parts of s). Sub in:σ iω αe sσ e isω αe sσ [cos(ωT ) i sin(ωT )]Take real and imaginary parts:σ αe σT cos(ωT )ω αe σT sin(ωT )real partimaginary partSeek a solution with σ 0. Things simplify quite a bit:0 α cos(ωT )ω α sin(ωT )Squaring and adding gives ω 2 α2 so ω α. This is not too surprising: we shouldexpect complex conjugate pairs of solutions. Could limit calculuations to ω 0 ifit helped, and just remember the complex conjugates are also there. In any case,subbing in either solution to the second equation, gives the same outcome:9
sin(αT ) 1so αT π 5π 9π, , ,.2 2 2Copyright c 2019 University of Cambridge. Not to be quoted or reproduced without permissionand no need to worry about negative solutions, as both α 0 and T 0. So thinkingabout increasing T up from zero, we have a complex root switch real part sign manytimes. We are interested in the first one: αT π/2.This is optional, but as this is the first example, we will check that we really do haveπ. It turns out to be sensible to split into two casesstable solutions when 0 T 2αaccording to modulus of ω For ω α, from considering modulus in the equation for the real part we seethat exp( σT ) 1 hence σ 0. For ω α, ωT αT π/2 so cos(ωT ) 0. From the equation for the realpart, we see σ 0 again.So either way, we have negative σ and hence stable solutions. Note we have notactually found any values for s, but we have shown they will have negative real part inthis range.1Numerical simulation is consistent with 0 T αesolutions decay exponentially to1πthe fixed point; for αe T 2α solutions decay and oscillate to the fixed point, and forπthe solution is unstable and heads to a cycle. This is typical: delay-differentialT 2αequation models often lead to oscillatory solutions.T 0.3 α 1 3.03.02.52.52.02.01.51.51.01.00.50.00.50 10203040 0.0T 1.55 α 10103.02.52.52.02.01.51.51.01.00.52030403040 T 1.8 α 1 3.00.0T 1.2 α 1 0.5010203040 0.001020(See Mathematica: Delay Logistic Equation)10
Under the carpetTreat this note as starred. If you are happy with DDEs already, then skip it. If you areconcerned that something might have been swept under the carpet here, you are right,so read on. What we have actually done isCopyright c 2019 University of Cambridge. Not to be quoted or reproduced without permission Found where there are real solutions for s and shown they are negative. For first range of T showed that any solution for s has negative real part Found all the values of T 0 where a solution has real part zeroWe can actually know more about the solutions for s of sT esT αT if we read up on‘Lambert W-Functions’. There are many solutions. Mathematica has a built-in functionthat can be used to give them numerically, and we can plot their real part as a functionof T (set α 1 for π23π7π24π9π25πT-0.10-3.0σ Re(s) against T for α 1 for top 10 solutions(same plot each side, just different vertical scale)The root with the largest real part (top line on graph) actually corresponds to that largestreal solution to start with, and you can see the sharp change of direction as it becomescomplex. The vertical zoom-in on the right shows more clearly that successive (pairsof) solutions pass upwards into positive σ.For T 0 we only had one value of s (namely s α). This was enough to determinethe linear behaviour of any small perturbation: 0 exp ( αt), and we’d just need toput in the appropriate single constant 0 . Now, for a perturbation, we should specify notjust (0) but also (t) for the interval t [ T, 0]. And the resultant dynamics will be asa sum of these types of solution with different s: (t) Xi11ai esi t
where the ai are determined by the initial conditions. The si with the largest real partwill end up dominating as t increases.Copyright c 2019 University of Cambridge. Not to be quoted or reproduced without permissionSo, actually we have got the dynamics right from the simple approach we first took: thereal solution dominated when it existed, the we had an oscillatory but decaying solutionuntil we found the lowest T where things could lose stability. For math bio, treat thissimple approach as sufficient.Be a bit careful when rescaling DDEsThis is just a word of caution about rescaling delay differential equations with respectto time. In short, you must remember to rescale any time lag also. In long, we will usethe above as an example:d (t) α (t T )dtThere are two parameters here, α and T . It is tempting to try and get rid of α byrescaling time. We set t̂ αt to cancel out with the α:d (t)d α α (t T )dtdt̂sodη(t̂) (t T ) η(α(t T )) η(t̂ αT )dt̂and finally:dη(t̂) η(t̂ αT ).dt̂So really we have not eliminated α but we have compounded our two parameters to asingle parameter combination αT .In general, be aware that the lag needs to rescale with time also. It is not usual inpractice to write out all of these steps. It is usually acceptable to reuse the originalvariable name ( here), but the change was made explicit just this once.Exercise 6: Find the equivalent of equation (1) for this rescaled DDE. (It turnsout to be slightly different, but it ought to give us the same conditions for stability.)12
DDE Example: BlowfliesCopyright c 2019 University of Cambridge. Not to be quoted or reproduced without permissionThis example stems from classic experimental work by Nicholson and others in the1950s on the Australian sheep blowfly. Populations of flies were kept in the lab andpopulation size was tracked over time, showing some quite spectacular fluctuationsdespite the available food and other external factors being kept steady. The full lifecycle of these flies is a few weeks (eggs, larval stages, then adult). Mathematicalbiologists have modelled this using delay differential equations.The unusual thing here is that the number of eggs produced
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