|Dr. Marcus R Garvie
- Blowey J.F. and Garvie M.R.
(2005), "A reaction-diffusion system of lambda-omega type. Part I: mathematical analysis,"
published in European Journal of Applied Mathematics, Vol. 16 No. 01 , pp. 1-19
We rigorously establish the well-posedness of the strong solutions for
an important class of oscillatory reaction-diffusion systems containing
a supercritical Hopf bifurcation in the reaction kinetics. This is
achieved using the classical Faedo-Galerkin method of Lions and
compactness arguments. Furthermore, we present a complete case study
for the application of this method to systems of reaction-diffusion
- Garvie M.R. and Blowey J.F. (2005), "A reaction-diffusion system of
lambda-omega type. Part II: numerical analysis," published in European Journal of Applied Mathematics, Vol. 16 No. 05
, pp. 621-646
In this paper the results from the Part I paper are mimicked in the discrete case
to present results for a fully-practical piecewise linear finite element method.
A priori estimates and error bounds are established in the semi-discrete and the
fully-discrete cases. The theoretical results are also illustrated in 1-D and 2-D.
In the 1-D case solutions are typically periodic travelling waves, while in 2-D
solutions may be spiral waves or 'target patterns'.
- Garvie M.R. and Trenchea C.
(2007), "Finite element approximations of spatially extended predator-prey interactions with the Holling
type II functional response," published in Numerische Mathematik, Vol. 107, pp. 641-667.
We present the numerical analysis of two well-known reaction-diffusion
systems modeling predator-prey interactions, where the local growth of prey is
logistic and the predator displays the Holling type II functional
response. Results are presented for two
fully-practical piecewise linear finite element methods. We establish
a priori estimates and error bounds for the semi-discrete and
fully-discrete finite element approximations. Numerical results
illustrating the theoretical results and spatiotemporal phenomena
(e.g., spiral waves and chaos) are presented in 1-D and 2-D.
- Garvie M.R. (2007)
, "Finite difference schemes for reaction-diffusion equations
modeling predator-prey interactions in MATLAB," published in
Bulletin of Mathematical Biology, Vol. 69. No. 3, pp. 931-956.
We present two finite difference algorithms for studying the dynamics of
spatially extended predator-prey interactions with the Holling Type
II functional response and logistic growth of the prey. The algorithms are stable and
convergent provided the time step is below a (non-restrictive)
critical value. This is advantageous as it is well-known that the
dynamics of nonlinear differential equations (DEs) can differ
significantly from that of the underlying DEs themselves. This is
particularly important for the spatially extended systems that are
studied in this paper as they display a wide spectrum of
ecologically relevant behavior, including chaos.
Furthermore, there are implementational advantages
of the methods. For example, due to the structure of the resulting linear
systems, standard direct and iterative solvers are guaranteed to
converge. Thus the algorithms are ideal for investigating the spatiotemporal
dynamics of the solutions. We also present the results of numerical
experiments in one and two space dimensions, and illustrate the
simplicity of the numerical methods with short programs in
MATLAB. Readers can download and edit the codes from
PRED_PREY_SIM: Predator-Prey codes in Matlab.
- Garvie M.R. and Trenchea C. (2007)
, "Optimal control of a
'nutrient-phytoplankton-zooplankton-fish' system," published in
SIAM Journal On Control and Optimization, Vol. 46 No. 3, pp. 775-791.
We consider the mathematical formulation, analysis, and numerical
solution of an optimal
control problem for a nonlinear `nutrient-phytoplankton-zooplankton-fish'
reaction-diffusion system. We study the existence of optimal
solutions, derive an optimality system, and determine optimal
solutions. In the original spatially homogeneous
formulation the dynamics of plankton were investigated as
a function of parameters for nutrient levels and fish predation rate
on zooplankton. In our paper the model is spatially extended and
the parameter for fish predation treated as a control variable. The
model has implications for the biomanipulation of food-webs in
eutrophic lakes to help improve water quality. In order to illustrate
the control of irregular spatiotemporal dynamics of plankton in
the model we implement a semi-implicit (in time) finite element method
with 'mass lumping', and present the results of numerical experiments
in two space dimensions.
- Garvie M.R. and Golinski M. (2010)
, "Metapopulation dynamics for spatially-extended
predator-prey interactions," published in
Ecological Complexity, Vol. 7 No. 2, pp. 55-59.
Traditional metapopulation theory classifies a metapopulation as a spatially
homogeneous population that persists on neighboring habitat patches. The
fate of each population on a habitat patch is a function of a balance between
births and deaths via establishment of new populations through migration to
neighboring patches. In this study, we expand upon traditional metapopulation
models by incorporating spatial heterogeneity into a previously studied two-patch
nonlinear ordinary differential equation metapopulation model, in which the
growth of a general prey species is logistic and growth of a general predator
species displays a Holling type II functional response. The model described
in this work assumes that migration by generalist predator and prey populations
between habitat patches occurs via a migratory corridor. Thus, persistence of
species is a function of local population dynamics and migration between
spatially heterogeneous habitat patches. Numerical results generated by our model
demonstrate that population densities exhibit periodic plane-wave phenomena,
which appear to be functions of differences in migration rates between generalist
predator and prey populations. We compare results generated from our model to
results generated by similar, but less ecologically realistic work, and to
observed population dynamics in natural metapopulations.
- Garvie M.R. and Trenchea C.
(2010), "Spatiotemporal dynamics of two generic predator-prey models,"
published in Journal of Biological Dynamics, Vol. 4 No. 6, pp. 559-570.
We present the analysis of two reaction-diffusion systems modeling
predator-prey interactions with the Holling Type II functional
response and logistic growth of the prey. Initially we undertake the
local analysis of the systems, deriving conditions on the parameters
that guarantee a stable limit cycle in the reaction kinetics, and construct
arbitrary large invariant regions in the equal diffusion coefficient
case. We then provide an a priori estimate that leads to the
global well-posedness of the classical (nonnegative) solutions, given
any nonnegative L&infin- initial data. In order to verify the
theoretical results numerical results are provided in two space
dimensions using a Galerkin finite element method with piecewise
linear continuous basis functions.
- Schwalb A.N., Garvie M.R. and Ackerman J.D.
(2010), "Dispersion of freshwater mussel larvae in a
published in Limnology and Oceanography, Vol. 55 No. 2, pp. 628-638.
We examined the dispersal of larvae (glochidia) of a common unionid mussel
species Actinonaias ligamentina, which need to attach to a host fish in order
to develop into juveniles, in a lowland river (Sydenham River, Ontario, Canada).
Results from field trials are compared with the numerical solution of a 1D
convection-diffusion partial differential equation model.
- Garvie M.R., Maini P.K. and Trenchea C.
(2010), "An efficient and robust numerical algorithm for estimating parameters
in Turing systems," published in Journal of Computational Physics, Vol. 229, pp. 7058-7071
We present a new algorithm for estimating parameters in reaction-diffusion systems that display
pattern formation via the mechanism of diffusion-driven instability. A Modified Discrete Optimal
Control Algorithm (MDOCA) is illustrated with the Schnakenberg and Gierer-Meinhardt
reaction-diffusion systems using PDE constrained optimization techniques. The MDOCA algorithm is
a modification of a standard variable-step gradient algorithm that yields a huge saving in
computational cost. The results of numerical experiments demonstrate that the algorithm accurately
estimated key parameters associated with stationary target functions generated from the models
themselves. Furthermore, the robustness of the algorithm was verified by performing experiments
with target functions perturbed with various levels of additive noise. The MDOCA algorithm could
have important applications in the mathematical modeling of realistic Turing systems when
experimental data are available. A Modified Discrete Optimal Control Algorithm (MDOCA) is
illustrated with the Schnakenberg and Gierer-Meinhardt reaction-diffusion systems using PDE
constrained optimization techniques. The MDOCA algorithm is a modification of a standard
variable-step gradient algorithm that yields a huge saving in computational cost. The results of
numerical experiments demonstrate that the algorithm accurately estimated key parameters
associated with stationary target functions generated from the models themselves. Furthermore,
the robustness of the algorithm was verified by performing experiments with target functions
perturbed with various levels of additive noise. The MDOCA algorithm could have important
applications in the mathematical modeling of realistic Turing systems when experimental data
- Garvie M.R. and Trenchea C.
(2011), "A three level ﬁnite element approximation of a pattern
formation model in developmental biology,"
published in Numerische Mathematik (online first DOI 10.1007/s00211-013-0591-z).
This paper concerns a second-order, three level piecewise linear ﬁnite element scheme
2-SBDF [J. RUUTH, Implicit-explicit methods for reaction-diffusion problems in pattern formation,
J. Math. Biol., 34 (1995), pp. 148-176] for approximating the stationary (Turing) patterns of a well-
known experimental substrate-inhibition reaction-diffusion (‘Thomas’) system [D. THOMAS,
Artiﬁcial enzyme membranes, transport, memory and oscillatory phenomena, in Analysis and control of
immobilized enzyme systems, D. Thomas and J.P. Kernevez, eds., Springer, 1975, pp. 115-150]. A
numerical analysis of the semi-discrete in time approximations leads to semi-discrete a priori bounds
and an optimal error estimate. The analysis highlights the technical challenges in undertaking the
numerical analysis of multi-level (≥ 3) schemes. We illustrate the effectiveness of the numerical
method by repeating an important classical experiment in mathematical biology, namely, to
approximate the Turing patterns of the Thomas system over a schematic mammal skin domain with ﬁxed
geometry at various scales. We also make some comments on the correct procedure for simulating
Turing patterns in general reaction-diﬀusion systems.
- Garvie M.R. and Trenchea C.
(2011), "Identiﬁcation of space-time distributed parameters
in the Gierer-Meinhardt reaction-diﬀusion system "
SIAM Journal on Applied Mathematics, Vol. 74, No. 1, pp. 147-166
We consider parameter identiﬁcation for the classic Gierer-
Meinhardt reaction-diﬀusion system. The original Gierer-Meinhardt model [A.
Gierer and H. Meinhardt, Kybernetik, 12 (1972), pp. 30-39] was formulated with
constant parameters and has been used as a prototype system for investigating
pattern formation in developmental biology. In our paper the parameters are
extended in time and space and used as distributed control variables. The
methodology employs PDE-constrained optimization in the context of
image-driven spatiotemporal pattern formation. We prove the existence of optimal
solutions, derive an optimality system, and determine optimal solutions. The
results of numerical experiments in 2D are presented using the ﬁnite element
method, which illustrates the convergence of a variable-step gradient algorithm
for ﬁnding the optimal parameters of the system. A practical target function was
constructed for the optimal control algorithm corresponding to the actual image
of a marine angelﬁsh.
- Garvie M.R., Burkardt J. and Morgan J.
(2015), "Simple Finite Element Methods for Approximating Predator-Prey
Dynamics in Two Dimensions Using MATLAB "
Bulletin of Mathematical Biology, Vol. 77. No. 3, pp. 548-578
We describe simple finite element schemes for approximating spatially extended predator–prey dynamics
with the Holling type II functional response and logistic growth of the prey. The finite element
schemes generalize ‘Scheme 1’ in the paper by Garvie (Bull Math Biol 69(3):931–956, 2007). We present
user-friendly, open-source MATLAB code for implementing the finite element methods on arbitrary-shaped
two-dimensional domains with Dirichlet, Neumann, Robin, mixed Robin–Neumann, mixed Dirichlet–Neumann,
and Periodic boundary conditions. Users can download, edit, and run the codes from
http://www.uoguelph.ca/~mgarvie/. In addition to discussing the well posedness of the model equations,
the results of numerical experiments are presented and demonstrate the crucial role that habitat shape,
initial data, and the boundary conditions play in determining the spatiotemporal dynamics of predator–prey
interactions. As most previous works on this problem have focussed on square domains with standard boundary
conditions, our paper makes a significant contribution to the area.
- Garvie M.R. and Morgan J. and Sharma V.
(2017), "Finite element approximation of a spatially structured
metapopulation PDE model "
Computers and Mathematics with Applications, Vol. 74. No. 5, pp. 934–947
We present a new fully spatially structured PDE metapopulation model for predator–prey dynamics in d <= 3
space dimensions. A nonlinear reaction–diffusion system of Rosenzweig–MacArthur form models predator–prey
dynamics in two ‘high’ quality patches embedded in a ‘low’ quality subdomain, where species can diffuse,
convect and die. Our model substantially generalizes and improves earlier fully structured metapopulation
models. After a nondimensionalization procedure, in order to approximate the metapopulation model we
present a fully discrete Galerkin finite element method in two space dimensions, which is a generalization
of the finite element method analyzed in a previous single patch predator–prey model. The numerical
solutions are illustrated for some test cases using MATLAB. Numerical experiments demonstrate that the
initial local extinction of predators in one patch leads to waves of recolonization from another patch.
In an appendix we also give an outline for the proof of the well-posedness of the model.
- Diele F. and Garvie M.R. and Trenchea C.
Numerical analysis of a first-order in time implicit-symplectic scheme for predator-prey systems "
Computers and Mathematics with Applications, Vol. 74. No. 5, pp. 948-961
The numerical solution of reaction-diffusion systems modelling predator-prey dynamics using implicit-symplectic
(IMSP) schemes is relatively new. When applied to problems with chaotic dynamics they perform well,
both in terms of computational effort and accuracy. However, until the current paper, a rigorous numerical
analysis was lacking. We analyse the semi-discrete in time approximations of a first-order IMSP scheme applied
to spatially extended predator-prey systems. We rigorously establish semi-discrete a priori bounds that
guarantee positive and stable solutions, and prove an optimal a priori error estimate. This analysis is an
improvement on previous theoretical results using standard implicit-explicit (IMEX) schemes. The theoretical
results are illustrated via numerical experiments in one and two space dimensions using fully-discrete finite
Submitted papers: (links not active)
- Trenchea C. and Garvie M.R.
, "Biomanipulation of Food-Webs in Eutrophic Lakes.,"
published in Proceedings of the 46th IEEE Conference on Decision and Control,
New Orleans, Louisiana, USA.
This is an adaptation of the SIAM paper listed above for a conference presentation.