Mass Structured Systems with Boundary Delay: Oscillations and the Effect of Selective Predation

Abstract

We study equilibrium and oscillatory solutions of a general mass structured system with a boundary delay. Such delays may be derived from systems with a separate egg class. Analytical calculations reveal existence criteria for non-trivial steady states. We further explore parameter space using numerical methods. The analysis is applied to a typical mass structured slug population model revealing oscillations, pulse solutions and irregular dynamics. However, robustly defined isolated cohorts, of the form sometimes suggested by experimental data, do not naturally emerge. Nonetheless, disordered, leapfrogging local maxima do result and may be enhanced by selective predation.

Appendices

Appendix A: Hopf Bifurcation

Following on from Eq. (43), for a Hopf bifurcation, we put γ=iα, where α∈ℝ. For convenience, set

$$ h(x) = \int^{x}_{x_h}\frac{1}{g(\sigma)}\,\mathrm{d} \sigma , $$

(46)

so that G(x)=exp(h(x)). Then

$$ G^{\gamma}(x)=G^{\mathrm{i}\alpha}(x)=\cos\bigl(\alpha h(x)\bigr)+\mathrm{i}\sin \bigl(\alpha h(x)\bigr), $$

(47)

and

(48)

where we define

$$ K\bigl(x,M^*\bigr) = 1/\mu_M\bigl(x,M^*\bigr) $$

(49)

for notational convenience (see definition of carrying capacity in Sect. 3). Also we find that

(50)

Then, defining

(51)

(52)

and noting the definition of M ^∗, we obtain

(53)

(54)

Hence,

(55)

where a ₁₁=[CK(x,M ^∗)α+SM ^∗]/A and a ₁₂=[SK(x,M ^∗)α+S ²+C ²−CM ^∗]/A, where A=(K(x,M ^∗)α+S)²+(C−M ^∗)².

Finally, we obtain the two equations

(56)

that must be satisfied for the two unknowns α and the control parameter (yet to be given).

The starred quantities are known from calculation of the non-trivial equilibrium solution, and C and S are functions of α as defined in Eqs. (51) and (52), respectively. The function β(x) is the birth kernel, m(x) is the kernel for M, K(x,M ^∗) is a function from the death rate defined in Eq. (49), and h(x) is related to the known growth function, g(x), as in Eq. (46).

With the definitions in Sect. 3, a computation reveals that K(x,M ^∗) now simply equals the constant K ₀, and h(x) is computed from Eq. (46) to be $h(x)=g_{2} (\frac{1}{x_{m}-x}-\frac{1}{x_{m}-x_{h}} )$.

Appendix B: Numerical Scheme

The simulation of the model equations is made with a numerical method that integrates the equations along characteristic curves and uses a representation formula of the solution. See Bees et al. (2006) for full details. The method was further adapted to the new system with a boundary delay, which represents the egg stage, and employs a moving grid method with node selection as developed in Angulo and López-Marcos (2004). We briefly document the method here for completeness. For each time, the grid nodes and the approximations to the solution at these points are calculated by means of a characteristics method, except for slugs of minimum size, which are obtained by means of the boundary condition. The formula we use in the numerical method is based upon a theoretical integration along characteristics, which provides the next representation of the solution to problem (19). Hence,

$$ s\bigl(t,x(t;t_*,x_*)\bigr)=s(t_{*},x_{*}) \exp \biggl(-\int^{t}_{t_*} \mu^*\bigl(x( \tau;t_{*},x_{*}),M(\tau),\varPi(\tau)\bigr)\,\mathrm{d}\tau \biggr), $$

(57)

where μ ^∗(x,M(t),Π(t))=μ(x,M(t),Π(t))+g′(x), and x(t;t _∗,x _∗) is the solution of

$$ \left \{ \begin{array}{l} \dfrac{d x}{d t}=g\bigl(x(t)\bigr), \\[7pt] x(t_*)=x_*. \end{array} \right . $$

(58)

Given positive integers R and J with a fixed time interval [0,T], we define Δt=a _h /R, Δx=(x _m −x _h )/J and N=[T/Δt], with N+1 discrete time levels t _n =nΔt, 0≤n≤N. The initial grid nodes are chosen as $X_{j}^{0}=x_{h} + jh$, 0≤j≤R, with the numerical initial condition (equal to the theoretical initial condition at each node) $U_{j}^{0} = s(X^{0}_{j},0), 0\leq j \leq J$. The initial condition for a delay problem requires a value of the solution for [−a _h ,0]. However, the delay appears only at the boundary, so we store eggs laid for each time t, by introducing $L_{n} = \frac{x_{h}}{a_{h}} l(0, t_{n} )$, where l(0,t _n ) represents the eggs laid at t=t _n , −R≤n≤0. For the general time step, t _n+1, 0≤n≤N−1, we assume that the solution approximations and grid at the previous time level t _n are known. Then, defining

(59)

(60)

(61)

(62)

we compute

$$ U^{n+1}_0=\frac{L^{n-R+1}_0 \exp(-a_h \mu_0)}{g(x_h)},\quad\mbox{and}\quad L^{n+1}_0 = Q \bigl( \mathbf{X}^{n+1},\boldsymbol{\beta}^{n+1} \mathbf{U}^{{n+1}} \bigr). $$

(63)

Here, $\beta_{j}^{n}=\beta(X^{n}_{j})$, 0≤j≤J+1, $(\gamma_{s})_{j}^{n}=\gamma_{s}(X^{n}_{j})$, 0≤j≤J+1, s=M,Π; and $Q(\mathbf{ X}^{l},\mathbf{ V}^{l})= \sum_{j=0}^{J+l}q^{l}_{j}(\mathbf{X}^{l})V^{l}_{j}$, l=0,1, where $q^{l}_{j}(\mathbf{ X}^{l})$, 0≤j≤J+l, l=0,1, are the coefficients of the composite trapezoidal quadrature rules with nodes in X ^l. Also, $\boldsymbol{\gamma}^{n}_{s}\mathbf{U}^{n}$, s=M,Π; and β ⁿ U ⁿ, represent the component-wise product of the corresponding vectors, 0≤n≤N. Note that the functions γ _s , s=M,Π, are the kernels of the integrals in the definitions of functions M and Π.

The number of nodes might vary at consecutive time levels as new nodes are introduced to the scheme that flux through node x _h . Therefore, the first grid node $X^{n+1}_{l}$ that satisfies

$$ \bigl|X^{n+1}_{R+l+1}-X^{n+1}_{R+l-1}\bigr|= \min_{1 \leq j \leq J}\bigl|X^{n+1}_{R+j+1}-X^{n+1}_{R+j-1}\bigr|, $$

(64)

is removed, maintaining a constant number of nodes for each time level involved in the implementation of the step scheme ((J+2) and (J+1) for the current and previous levels, respectively). However, we do not recompute the approximations to the nonlocal terms at such time levels. The grid (subgrid of the complete system, in which all the nodes take part) is composed of x _h , the minimum size of slugs, together with the nodes obtained by integration along characteristics from the nodes selected at the previous time level. The solutions are defined by an implicit system of equations given by (61), which are solved via an iterative procedure. (This scheme can be easily extended to treat problems where the slug growth function depends additionally on a weighted sum of the population.)