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Introduction to Mathematical Biology(G5106)Lecture notes 2020-Dr Yuliya KyrychkoDepartment of MathematicsUniversity of Sussex, UKContents 1 Continuous population models for single species 1 Basic Conservation Law 1.1 Malthus and Verhulst models 1.1 Stationary points 1.1 Linear stability analysis 1 I...

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Introduction to Mathematical Biology (G5106)

Lecture notes 2020-2021 Dr Yuliya Kyrychko

Department of Mathematics University of Sussex, UK

Contents 1 Continuous population models for single species 1.1 Basic Conservation Law . . . . . . . . . . . . . . 1.1.1 Malthus and Verhulst models . . . . . . . 1.1.2 Stationary points . . . . . . . . . . . . . . 1.1.3 Linear stability analysis . . . . . . . . . . 1.2 Insect outbreak model (Spruce Budworm) . . . . 1.2.1 Non-dimensalisation . . . . . . . . . . . . 1.2.2 Steady states . . . . . . . . . . . . . . . . 1.3 Harvesting a single natural population . . . . . . 1.3.1 Model with harvesting: constant yield . . 1.3.2 Model with harvesting: constant effort . . 1.3.3 Recovery time for model with harvesting .

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2 Discrete population models for single species 2.1 Main examples of the discrete models . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Linear stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Some general properties of the eigenvalues . . . . . . . . . . . . . . . . . 2.2 Bifurcation analysis of a discrete population model . . . . . . . . . . . . . . . . 2.2.1 Linear stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Iterative procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Phase plane analysis 3.1 Second order systems of ordinary differential equations (ODEs) . . . . . . . . . 3.1.1 Phase portrait properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Steady states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Classification of the fixed points: real eigenvalues . . . . . . . . . . . . . . . . . 3.2.1 Node, Type I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Saddle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Classification of the fixed points: complex eigenvalues . . . . . . . . . . . . . . . 3.4 Classification of the fixed points: equal eigenvalues . . . . . . . . . . . . . . . .

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4 Interacting populations (continuous models) 4.1 Predator-prey models . . . . . . . . . . . . . . 4.1.1 Stationary points and their stability . . 4.1.2 Integral curves . . . . . . . . . . . . . 4.2 Finite predation . . . . . . . . . . . . . . . . . 4.2.1 Alphids-Ladybirds (two-species) model

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4.2.2 Global behaviour at a glance: Nullclines 4.3 Competitive exclusion . . . . . . . . . . . . . . 4.3.1 Steady states . . . . . . . . . . . . . . . 4.4 Mutualism (symbiosis) . . . . . . . . . . . . . .

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5 Enzyme kinetics 5.1 Michaelis-Menten reaction . . . . . . . . . . . . . . . 5.1.1 Non-dimensionalisation . . . . . . . . . . . . . 5.1.2 Michaelis-Menten quasi-steady state analysis . 5.2 Enzyme Inhibition . . . . . . . . . . . . . . . . . . . 5.2.1 Systems of enzymes . . . . . . . . . . . . . . . 5.2.2 Competitive Inhibition . . . . . . . . . . . . . 5.3 Cooperativity . . . . . . . . . . . . . . . . . . . . . .

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6 Dynamics of infectious diseases and epidemics 6.1 Derivation of SI and SIS Models . . . . . . . . . . 6.1.1 SI model . . . . . . . . . . . . . . . . . . . 6.1.2 SIS model . . . . . . . . . . . . . . . . . . 6.2 Kermack-McKendrick (SIR) model . . . . . . . . 6.3 Modelling of sexually transmitted infections (STI) 6.4 Modelling of measles epidemic . . . . . . . . . . .

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List of Figures 1.1 Sketch illustrating asympotic behaviour of the function N (t) = N0 ert as t −→ ∞. N0 Kert 1.2 Sketch illustrating the asympotic behavior of the function N (t) = K + N0 (ert − 1) as t −→ ∞. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K K < N0 < K. . . . . . . . . . . . . and 1.3 The qualitative difference for N0 < 2 2 ′ ∗ 1.4 Picture illustrating that since f (N ) < 0 for N ∗ = 0, 5, the steady states N ∗ = 0 and N ∗ = 5 are stable, while since f ′ (N ∗ ) > 0 for N ∗ = 3, 7, the steady states N ∗ = 3 and N ∗ = 7 are unstable. . . . . . . . . . . . . . . . . . . . . . . . . . . BN 2 as a function of N . . . . . . . . . . . . . . . . . . . . 1.5 A plot of P (N ) = 2 A + N2 1.6 Plots of h(u) and g(u) with three intersection points u1 , u2 and u3 . . . . . . . . 1.7 Plots of h(u) and g(u) with just one intersection point u1 . . . . . . . . . . . . . 1.8 (a) Parameter domain shows the regions of one or three non-zero steady states for the budworm model. The region for three non-zero steady states is given by √ 2a3 2a3 the functions r(a) = 2 for a ≥ 3 giving the value of and q(a) = 2 a −1 a +1 point P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 (a) Plot of the typical behaviour of function f (u) in the outer region of Figure 1.8, where only one non-trivial steady state u1 exists. (b) Plot of typical behaviour of function f (u) in the shaded (inner) region of Figure 1.8, where there are three non-zero steady state u1 , u2 and u3 . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Constant yield harvesting model for different values of Y0 . . . . . . . . . . . . . 1.11 (a) Graphical method for finding the steady state for the harvesting logistic model. Logistic growth curve rN (1 − N/K) and yield Y = EN for two different values of E (higher value of E is represented by the dashed curve). (b) Ratio of plotted against YY from (1.12). I+ branch represents the recovery times, TTRR(E) (0) M the positive root (solid line) and I− represents the negative root (dashed line). . 2.1 A typical growth form of Nt+1 = f (Nt ). . . . . . . . . . . . . . . . . . . . . . . . 2.2 (a) Graphical method of finding the steady state N ∗ < Nm as a point of crossing of Nt+1 = f (Nt ) with Nt+1 = N1 . Cobwebbing technique is shown with arrows. . 2.3 (a) Graphical method of finding the steady state N ∗ > Nm as a point of crossing of Nt+1 = f (Nt ) with Nt+1 = N1 . Cobwebbing technique is shown as dashed lines. (b) Time evolution of population growth; continuous curve is used for illustration only — the population may change abruptly at each time step. . . . 2.4 Case 2.1, −1 < f ′ (N ∗ ) < 0, N ∗ is stable. . . . . . . . . . . . . . . . . . . . . . . 2.5 Case 2.2, f ′ (N ∗ ) = −1, N ∗ is neutrally stable (periodic oscillations around N ∗ ). 2.6 Case 2.3, f ′ (N ∗ ) < −1, N ∗ is unstable. . . . . . . . . . . . . . . . . . . . . . . . 3

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2.7 Ricker discrete population model (2.2) with N0 = 5 and K = 80 in both figures. (a) r = 1.5. (b) r = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Bifurcation diagram for the non-dimensonal discrete logistic model (2.6). Nonzero steady state is given by u∗ = (r − 1)/r for r > 1. Solid lines are stable steady states and dashed lines are unstable steady states. . . . . . . . . . . . . . 2.9 Second iteration ut+2 vs ut of the non-dimensional discrete logistic model. (a) r = 2.2; (b) r = 3.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Bifurcation diagram of the iterative logistic model, which illustrates changes in stability of the steady states as r goes through the critical values (pitchfork bifurcation). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Bifurcation diagram of the iterative logistic model, which illustrates chaotic behaviour. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

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Phase portrait of a stable node: all trajectories go to (0, 0). . . . . . . . . Phase portrait of a saddle point. . . . . . . . . . . . . . . . . . . . . . . . Phase portrait of an unstable spiral. . . . . . . . . . . . . . . . . . . . . . Phase portrait of a stable spiral. . . . . . . . . . . . . . . . . . . . . . . . Phase portrait of a center point singularity. . . . . . . . . . . . . . . . . . Phase portrait of an unstable node (type II), λ > 0. . . . . . . . . . . . . Phase portrait of a stable star, λ < 0. . . . . . . . . . . . . . . . . . . . . Summary diagramme showing different types of fixed point singularities.

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4.1 Dynamics of Lotka-Volterra model (4.1) for α = 1.095 and different values of constant C. (a) Phase portrait in (u, v)-plane. (b) Temporal evolution of u and v. 49 4.2 Examples of predator response N R(N ) to prey density. (a) R(N ) = A, the unsaturated Lotka-Volterra type. (b) R(N ) = A/(N +B). (c) R(N ) = AN/(N 2 +B 2 ). (d) R(N ) = A(1 − exp(−aN ))/N . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.3 The u−v phase plane for the Alphids-Ladybirds model when the non-zero steady state is stable. The u nullclines are plotted in green and v nullclines are plotted in red. Phase trajectories are plotted for several initial conditions as black curves. 54 4.4 Dynamics of the competitive exclusion model. The u nullclines are plotted in red and v nullclines are plotted in green. (a) a12 = 0.8 < 1, a21 = 1.2 > 1. (b) a12 = 1.2 > 1, a21 = 0.8 < 1. (c) a12 = 1.2 > 1, a21 = 1.2 > 1. (d) a12 = 0.8 < 1, a21 = 0.8 < 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.5 Dynamics of the mutualism model. The u nullclines are plotted in red and v nullclines are plotted in green. (a) a12 = 1.2 > 1, a21 = 1.2 > 1. (b) a12 = 0.2 < 1, a21 = 0.2 < 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.1 Phase trajectories in the susceptible (S)-infectives (I) phase plane for the SIR epidemic model (6.5). The curves are determined by the initial conditions I(0) = I0 and S(0) = S0 . With R(0) = 0, all trajectories start on the line S + I = N and remain within the triangle since 0 < S + I < N for all time. An epidemic exists if I(t) > I0 for any time t > 0, this always occurs if S0 > ρ and I0 > 0 . . . 74 6.2 Criss-cross SIR model. In this model, I ∗ infects S and I infects S ∗ . . . . . . . . 75 6.3 Criss-cross SI model. In this model, I ∗ infects S and I infects S ∗ , and after recovery individuals become susceptible straighaway. . . . . . . . . . . . . . . . 76

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Introduction to Mathematical Biology (G5106) 2020-21

Lecture notes

Chapter 1 Continuous population models for single species 1.1

Basic Conservation Law

The main aspect of population models is the conservation of population and the conservation equations for the population can be written as

rate of change of population = births − deaths + migration.

(1.1)

Let N (t) be the size of the population at time t. For simplicity, we will assume that the system is closed. This means that there is no migration, i.e. no-one enters population from the outside and no-one leaves the population (think of an isolated population on an island, where no other population is allowed to enter the island and the inhabitants cannot leave the island). Under this assumption, equation (1.1) becomes dN = births − deaths = f (N ) = N (t)g (N (t)), dt where g(N ) is the intrinsic growth rate.

(1.2)

Two main examples of such models are Malthus model and Verhulst model, which are described below.

1.1.1

Malthus and Verhulst models

Malthus Model (1978) Let g(N ) = births − deaths = b − d = r,

where b and d are constant birth and death rates, respectively. With this function g(N ) equation (1.2) becomes dN = r N, given N (0) = N0 > 0, dt Dr Yuliya Kyrychko

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where N0 is the initial population at time 0. Solving this differential equation yields

Figure 1.1: Sketch illustrating asympotic behaviour of the function N (t) = N0 ert as t −→ ∞. N (t) = N0 er t ≡ N0 e(b−d) t .

Thus if b > d the population grows exponentially and if b < d it dies out, i.e.   ∞ if r > 0, N (t) = N0 ert −→  0 if r < 0.

Verhulst Model (1836)

This model is also called the logistic growth model with function g(N ) defined as follows,   N g(N ) = r 1 − , K

i.e. as N increases, the intrinsic growth rate (per capita growth rate) decreases due to overcrowding, lack of food and competition.

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Definition 1 In the logistic growth equation, the constant r is the linear growth rate and the constant K is the carrying capacity.

Therefore, for N ≪ K, one has dN ≃ rN =⇒ N ≃ N0 ert . dt It can also be observed that as N → K , dN → 0. dt The logistic growth equation has the form dN = rN dt

  N 1− . K

(1.3)

Figure 1.2: Sketch illustrating the asympotic behavior of the function N (t) = as t −→ ∞.

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N0 Kert K + N0 (ert − 1)

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Solving the last equation with the initial condition N (0) = N0 gives N (t) =

N0 Kert −→ K, K + N0 (ert − 1)

as t → ∞

The behaviour of function N (t) as t → ∞ is illustrated in Figure 1.2. The logistic growth model has been shown to be a very good approximation for fitting population data from bacteria to yeast to rats and sheep. There is a qualitative difference for N0 <

K 2

and

K 2

< N0 < K, which is shown in Figure 1.3.

Figure 1.3: The qualitative difference for N0 <

1.1.2

K K and < N0 < K. 2 2

Stationary points

Consider a general population model, which can be written as dN = f (N ), dt where f (N ) is generally a nonlinear function of N .

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(1.4)

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Lecture notes

Definition 2 (Steady States) Fixed points, or equilibria, or steady states are stationary points of the systems, where the dynamics does not change in time.

The steady states of system (1.4) are constants N ∗ , which can be found as roots of f (N ) = 0. Example. If f (N ) = N (N − 3)(N − 5)(N − 7), there are four steady states, namely, N = 0, 3, 5, 7. Fixed points can be stable or unstable.

Definition 3 (Stability) A fixed point is stable if a small perturbation from that fixed point decays to zero, i.e. solution returns to the fixed point. A fixed point is unstable if a small perturbation grows exponentially and solution moves away from that fixed point.

1.1.3

Linear stability analysis

If we want to determine the stability of any steady state, we can perform a linear stability analysis. Let N ∗ be a steady state of the system dN/dt = f (N ) and take a small perturbation in the form: N (t) = N ∗ + n(t),

where n(t) ≪ 1.

Using Taylor expansion of f (N ) gives   ∂ 2 f  ∂f  1 2 + .... f (N ) = f (N + n(t)) = f (N ) + n + n (t) ∂N N =N ∗ 2 ∂N 2 N =N ∗ ∗

∗

Substituting this expansion into the original equation leads to

dN dn = = f (N ) dt dt   ∂ 2 f  ∂f  1 2 ∗ + .... = f (N ) + n + n (t) ∂N N =N ∗ 2 ∂N 2 N =N ∗

Linearisation is obtained by neglecting the higher order terms and considering linear terms only, dn = f ′ (N ∗ )n(t). dt Dr Yuliya Kyrychko

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Since this equation is linear, we can find its solution as n(t) = n(t = 0) exp [f ′ (N ∗ )t] . We observe that as t goes to infinity  ∞ if f ′ (N ∗ ) > 0 N ∗ is unstable,  f ′ (N ∗ )t n(t) = n0 e −→  0 if f ′ (N ∗ ) < 0 N ∗ is linearly stable. Definition 4 (Linear stability) Let N ∗ be a steady state of system (1.4) and linearised solution near this steady state has the form ′

∗

n(t) = n0 ef (N )t . The steady state N ∗ is linearly stable if n(t) → 0 as t → ∞, that is if the following condition holds f ′ (N ∗ ) < 0, N ∗ is linearly stable.

Example. Determine the stability of the steady states of the following system dN = N (N − 3)(N − 5)(N − 7). dt Solution. There are four steady states N (N − 3)(N − 5)(N − 7) = 0 =⇒ N = 0, N = 3, N = 5, N = 7, i.e. N ∗ = 0, 3, 5, 7 and we can plot the graph of function f (N ) = N (N − 3)(N − 5)(N − 7), which is shown in the Figure 1.4. In order to determine the stability of these steady states, we have to calculate the sign of the first derivative of the function f (N ) at each of the steady states. Differentiating f (N ) with respect to N gives, f ′ (N ) = 4N 3 − 45N 2 + 142N − 105.

Evaluating the derivative at each of the steady states, we obtain

f ′ (0) = −105 < 0 =⇒ N ∗ = 0 is stable, Dr Yuliya Kyrychko

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Figure 1.4: Picture illustrating that since f ′ (N ∗ ) < 0 for N ∗ = 0, 5, the steady states N ∗ = 0 and N ∗ = 5 are stable, while since f ′ (N ∗ ) > 0 for N ∗ = 3, 7, the steady states N ∗ = 3 and N ∗...