Modelling biological invasions over homogeneous and inhomogeneous landscapes using level set methods

Abstract

The establishment, spread and subsequent degradation of existing environments by invasive species is a worldwide problem affecting native and agricultural ecosystems. The phenomenal cost to governments as a result of research and eradication or control drives the need to understand invasion characteristics. In this paper we develop a method for modelling the boundary of an invasion over time with model inputs being the initial distribution of the invasion and the speed at which the invasion front moves over time. This speed function can depend on the topography of the ground cover and we consider examples of homogeneous and inhomogeneous spread. The possibility of a long-distance dispersal event occurring is also considered. In particular, examples of the spread of emergent weeds and weeds which favour creeks and river beds in New Zealand are presented.

Appendices

Appendices

Appendix A: Formulation of a partial differential equation for the level set function ϕ(x,y,t)

We begin with the level curve specified by the implicit function

$$ \phi(x,y,t)=0. $$

(13)

A point moving with the zero level curve can be described by x and y which can be functions of t, i.e. x = x(t) and y = y(t). Differentiation of equation (13) with respect to t gives

$$ \frac{\partial\phi}{\partial t}\frac{dt}{dt} +\frac{\partial \phi}{\partial x}\frac{dx}{dt} +\frac{\partial \phi}{\partial y}\frac{dy}{dt}=0. $$

(14)

Using vector notation equation (14) becomes

$$ \frac{\partial \phi}{\partial t} + \frac{d{\mathbf{x}}}{dt}\cdot\nabla \phi =0, $$

(15)

where

$$ {\mathbf{x}}=[x,y] \qquad \hbox{and }\qquad\,\nabla \phi=\left[\frac{\partial\phi}{\partial x},\frac{\partial\phi}{\partial y}\right]. $$

(16)

Suppose that at any particular time t, a point on the zero level curve at grid location (x,y) moves with speed v(x,y,t) in the direction of the normal to the curve, i.e. in a direction perpendicular to the tangent to the curve at that point. If v > 0 then the direction is outward; if v < 0 then the direction is inward. The unit normal to the function ϕ(x,y,t) at the point (x,y) is

$$ {\mathbf{n}}=\frac{\nabla \phi}{|\nabla \phi|}, $$

(17)

where

$$ |\nabla\phi|=\sqrt{\left(\frac{\partial\phi}{\partial x} \right)^2+\left(\frac{\partial\phi}{\partial y}\right)^2}. $$

(18)

Then

$$ \frac{d{\mathbf{x}}}{dt}=v(x,y,t){\mathbf{n}}. $$

(19)

Substitution into equation (15) gives

$$ \phi_t+v(x,y,t)\sqrt{\left(\frac{\partial\phi}{\partial x}\right)^2+ \left(\frac{\partial\phi}{\partial y}\right)^2} =0. $$

(20)

Appendix B: Numerical method for solving the level set equation

This section describes a stable numerical method for solving the level set Eq. 1. The method uses upwind finite differences to ensure that information propagates in the direction of movement of the zero level curve (Sethian 1996). The domain is discretised as x ₁ , x ₂ , … x _i … x _I , y ₁, y ₂, … y _j … y _J and t ₁, t ₂, … t _k … t _K . The numerical derivative of ϕ with respect to t can be approximated using a forward finite difference i.e.

$$ \phi_t\approx\frac{\phi(x_i,y_j,t_{k+1})-\phi(x_i,y_j,t_k)}{\triangle t}. $$

(21)

The function value of ϕ at location (x _i ,y _j ) at any time t _k+1 is ϕ(x _i ,y _j ,t _k+1) and can be approximated from function values at the previous time step t _k according to the equation

$$ \phi(x_i,y_j,t_{k+1})=\phi(x_i,y_j,t_{k})+\triangle t v(x_i,y_j,t_k)\sqrt{\phi_x(x_i,y_j,t_k)^2+\phi_y(x_i,y_j,t_k)^2} $$

(22)

where ϕ _x and ϕ _y , the partial derivatives of ϕ with respect to x and y respectively, must be approximated using either forward or backward differences, depending on whether the curve is propagating outwards or inwards at that point. A forward difference approximation of ϕ _x is denoted D ⁺ _x where

$$ \phi_x \approx D_x^+ =\frac{\phi(x_{i+1},y_j,t_k)-\phi(x_i,y_j,t_k)}{\triangle x}. $$

(23)

A backward difference approximation is

$$ \phi_x\approx D_x^- =\frac{\phi(x_{i},y_j,t_k)-\phi(x_{i-1},y_j,t_{k})}{\triangle x}. $$

(24)

Similarly one can derive forward and backward approximations for ϕ _y as D ⁺ _y and D ⁻ _y respectively where

$$ \phi_y\approx D_y^+ =\frac{\phi(x_i,y_{j+1},t_k)-\phi(x_i,y_j,t_k)}{\triangle y} $$

(25)

and

$$ \phi_y \approx D_y^- =\frac{\phi(x_i,y_j,t_k)-\phi(x_i,y_{j-1},t_k)}{\triangle y}. $$

(26)

Recall that the direction in which the level curve is moving is given by the vector

$$ v{\mathbf{n}} = \left[ \begin{array}{c}\frac{v\phi_x}{|\phi|}\\ \frac{v\phi_y}{|\phi|} \end{array} \right]. $$

(27)

Thus we are interested in the signs of the components of this vector. If the speed at location (x,y) and time t is positive then we look at which direction the unit normal is heading according to Table 1. For example if ϕ _x <  0 and ϕ _y  > 0 (top-left cell of the table) then the first component of the velocity vector is negative and the second is positive, so the curve is moving in the direction of decreasing x and increasing y. Thus a forward difference should be used for ϕ _x and a backward difference for ϕ _y . This means that information at (x _i , y _j , t _k+1) is based on information at (x _i+1 ,y _j , t _k ), (x _i+1, y _j-1, t _k ) and (x _i , y _j-1, t _k ). Other cases are handled similarly as summarised in the table. Note that the table column and row headings can be written in terms of both the true partial derivatives (unknown) and their approximations. If v(x,y,t) < 0, the situation is the same but with the directions reversed in Table 1. All the possible cases for the signs of v, ϕ _x and ϕ _y can be conveniently represented as a single scheme by equations (2)–(8).

Table 1 When v is positive the direction of propagation of the level set curve ϕ(x,y,t) = 0 in time depends on the signs of ϕ _x and ϕ _y