A Mathematical Model for Biocontrol of the Invasive Weed Fallopia japonica.

Stephen A Gourley,Jing Li,Xingfu Zou

doi:10.1007/s11538-016-0195-8

Abstract

We propose a mathematical model for biocontrol of the invasive weed Fallopia japonica using one of its co-evolved natural enemies, the Japanese sap-sucking psyllid Aphalara itadori. This insect sucks the sap from the stems of the plant thereby weakening it. Its diet is highly specific to F. japonica. We consider a single isolated knotweed stand, the plant’s size being described by time-dependent variables for total stem and rhizome biomass. It is the larvae of A. itadori that damage the plant most, so the insect population is described in terms of variables for the numbers of larvae and adults, using a stage-structured modelling approach. The dynamics of the model depends mainly on a parameter h, which measures how long it takes for an insect to handle (digest) one unit of F. japonica stem biomass. If h is too large, then the model does not have a positive equilibrium and the plant biomass and insect numbers both grow together without bound, though at a lower rate than if the insects were absent. If h is sufficiently small, then the model possesses a positive equilibrium which appears to be locally stable. The results based on our model imply that satisfactory long-term control of the knotweed F. japonica using the insect A. itadori is only possible if the insect is able to consume and digest knotweed biomass sufficiently quickly; if it cannot, then the insect can only slow down the growth which is still unbounded.

Highlights

The Japanese knotweed Fallopia japonica has been present in the UK since 1825
The model we develop in this paper is for a single isolated knotweed stand, and the plant’s size is described by time-dependent variables for total stem and rhizome biomass
We have developed a model to study the bio-control of F. japonica using one of its co-evolved natural enemies, the Japanese sap-sucking psyllid A. itadori

Summary

Introduction

The Japanese knotweed Fallopia japonica has been present in the UK since 1825. It was originally introduced by the Victorians as an ornamental plant, but soon escaped from their gardens. Note that an individual stem can only accommodate a certain number of eggs so that intra-specific competitive effects apply at the level of the individual stem Due recognition of this fact is crucial, since an important aspect we wish to study is the possibility that the knotweed stand as a whole can grow without bound, and so can the number of larval and adult A. itadori. We exploit the fact that, for such a solution, as t → ∞ the variables L/S and A/S approach constants whose values are not immediately known but are determined later in terms of the growth rate λ This approximation linearises the four equations even though the function bp(·) is kept general. It is stressed that the foregoing analysis is purely formal

Equilibrium Population for Small h

Findings

Discussion