Modelling count data with partial differential equation models in biology.

Partial differential equation (PDE) models are often used to study biological phenomena involving movement-birth-death processes, including ecological population dynamics and the invasion of populations of biological cells. Count data, by definition, is non-negative, and count data relating to biological populations is often bounded above by some carrying capacity that arises through biological competition for space or nutrients. Parameter estimation, parameter identifiability, and making model predictions usually involves working with a measurement error model that explicitly relating experimental measurements with the solution of a mathematical model. In many biological applications, a typical approach is to assume the data are normally distributed about the solution of the mathematical model. Despite the widespread use of the standard additive Gaussian measurement error model, the assumptions inherent in this approach are rarely explicitly considered or compared with other options. Here, we interpret scratch assay data, involving migration, proliferation and delays in a population of cancer cells using a reaction-diffusion PDE model. We consider relating experimental measurements to the PDE solution using a standard additive Gaussian measurement error model alongside a comparison to a more biologically realistic binomial measurement error model. While estimates of model parameters are relatively insensitive to the choice of measurement error model, model predictions for data realisations are very sensitive. The standard additive Gaussian measurement error model leads to biologically inconsistent predictions, such as negative counts and counts that exceed the carrying capacity across a relatively large spatial region within the experiment. Furthermore, the standard additive Gaussian measurement error model requires estimating an additional parameter compared to the binomial measurement error model. In contrast, the binomial measurement error model leads to biologically plausible predictions and is simpler to implement. We provide open source Julia software on GitHub to replicate all calculations in this work, and we explain how to generalise our approach to deal with coupled PDE models with several dependent variables through a multinomial measurement error model, as well as pointing out other potential generalisations by linking our work with established practices in the field of generalised linear models.

Geometric analysis enables biological insight from complex non-identifiable models using simple surrogates.

An enduring challenge in computational biology is to balance data quality and quantity with model complexity. Tools such as identifiability analysis and information criterion have been developed to harmonise this juxtaposition, yet cannot always resolve the mismatch between available data and the granularity required in mathematical models to answer important biological questions. Often, it is only simple phenomenological models, such as the logistic and Gompertz growth models, that are identifiable from standard experimental measurements. To draw insights from complex, non-identifiable models that incorporate key biological mechanisms of interest, we study the geometry of a map in parameter space from the complex model to a simple, identifiable, surrogate model. By studying how non-identifiable parameters in the complex model quantitatively relate to identifiable parameters in surrogate, we introduce and exploit a layer of interpretation between the set of non-identifiable parameters and the goodness-of-fit metric or likelihood studied in typical identifiability analysis. We demonstrate our approach by analysing a hierarchy of mathematical models for multicellular tumour spheroid growth experiments. Typical data from tumour spheroid experiments are limited and noisy, and corresponding mathematical models are very often made arbitrarily complex. Our geometric approach is able to predict non-identifiabilities, classify non-identifiable parameter spaces into identifiable parameter combinations that relate to features in the data characterised by parameters in a surrogate model, and overall provide additional biological insight from complex non-identifiable models.

Profile-Wise Analysis: A profile likelihood-based workflow for identifiability analysis, estimation, and prediction with mechanistic mathematical models.

Interpreting data using mechanistic mathematical models provides a foundation for discovery and decision-making in all areas of science and engineering. Developing mechanistic insight by combining mathematical models and experimental data is especially critical in mathematical biology as new data and new types of data are collected and reported. Key steps in using mechanistic mathematical models to interpret data include: (i) identifiability analysis; (ii) parameter estimation; and (iii) model prediction. Here we present a systematic, computationally-efficient workflow we call Profile-Wise Analysis (PWA) that addresses all three steps in a unified way. Recently-developed methods for constructing 'profile-wise' prediction intervals enable this workflow and provide the central linkage between different workflow components. These methods propagate profile-likelihood-based confidence sets for model parameters to predictions in a way that isolates how different parameter combinations affect model predictions. We show how to extend these profile-wise prediction intervals to two-dimensional interest parameters. We then demonstrate how to combine profile-wise prediction confidence sets to give an overall prediction confidence set that approximates the full likelihood-based prediction confidence set well. Our three case studies illustrate practical aspects of the workflow, focusing on ordinary differential equation (ODE) mechanistic models with both Gaussian and non-Gaussian noise models. While the case studies focus on ODE-based models, the workflow applies to other classes of mathematical models, including partial differential equations and simulation-based stochastic models. Open-source software on GitHub can be used to replicate the case studies.

Growth and adaptation mechanisms of tumour spheroids with time-dependent oxygen availability.

Tumours are subject to external environmental variability. However, in vitro tumour spheroid experiments, used to understand cancer progression and develop cancer therapies, have been routinely performed for the past fifty years in constant external environments. Furthermore, spheroids are typically grown in ambient atmospheric oxygen (normoxia), whereas most in vivo tumours exist in hypoxic environments. Therefore, there are clear discrepancies between in vitro and in vivo conditions. We explore these discrepancies by combining tools from experimental biology, mathematical modelling, and statistical uncertainty quantification. Focusing on oxygen variability to develop our framework, we reveal key biological mechanisms governing tumour spheroid growth. Growing spheroids in time-dependent conditions, we identify and quantify novel biological adaptation mechanisms, including unexpected necrotic core removal, and transient reversal of the tumour spheroid growth phases.

Making Predictions Using Poorly Identified Mathematical Models.

Many commonly used mathematical models in the field of mathematical biology involve challenges of parameter non-identifiability. Practical non-identifiability, where the quality and quantity of data does not provide sufficiently precise parameter estimates is often encountered, even with relatively simple models. In particular, the situation where some parameters are identifiable and others are not is often encountered. In this work we apply a recent likelihood-based workflow, called Profile-Wise Analysis (PWA), to non-identifiable models for the first time. The PWA workflow addresses identifiability, parameter estimation, and prediction in a unified framework that is simple to implement and interpret. Previous implementations of the workflow have dealt with idealised identifiable problems only. In this study we illustrate how the PWA workflow can be applied to both structurally non-identifiable and practically non-identifiable models in the context of simple population growth models. Dealing with simple mathematical models allows us to present the PWA workflow in a didactic, self-contained document that can be studied together with relatively straightforward Julia code provided on GitHub . Working with simple mathematical models allows the PWA workflow prediction intervals to be compared with gold standard full likelihood prediction intervals. Together, our examples illustrate how the PWA workflow provides us with a systematic way of dealing with non-identifiability, especially compared to other approaches, such as seeking ad hoc parameter combinations, or simply setting parameter values to some arbitrary default value. Importantly, we show that the PWA workflow provides insight into the commonly-encountered situation where some parameters are identifiable and others are not, allowing us to explore how uncertainty in some parameters, and combinations of parameters, regardless of their identifiability status, influences model predictions in a way that is insightful and interpretable.

Computationally efficient mechanism discovery for cell invasion with uncertainty quantification.

Parameter estimation for mathematical models of biological processes is often difficult and depends significantly on the quality and quantity of available data. We introduce an efficient framework using Gaussian processes to discover mechanisms underlying delay, migration, and proliferation in a cell invasion experiment. Gaussian processes are leveraged with bootstrapping to provide uncertainty quantification for the mechanisms that drive the invasion process. Our framework is efficient, parallelisable, and can be applied to other biological problems. We illustrate our methods using a canonical scratch assay experiment, demonstrating how simply we can explore different functional forms and develop and test hypotheses about underlying mechanisms, such as whether delay is present. All code and data to reproduce this work are available at https://github.com/DanielVandH/EquationLearning.jl.

Efficient inference and identifiability analysis for differential equation models with random parameters.

Heterogeneity is a dominant factor in the behaviour of many biological processes. Despite this, it is common for mathematical and statistical analyses to ignore biological heterogeneity as a source of variability in experimental data. Therefore, methods for exploring the identifiability of models that explicitly incorporate heterogeneity through variability in model parameters are relatively underdeveloped. We develop a new likelihood-based framework, based on moment matching, for inference and identifiability analysis of differential equation models that capture biological heterogeneity through parameters that vary according to probability distributions. As our novel method is based on an approximate likelihood function, it is highly flexible; we demonstrate identifiability analysis using both a frequentist approach based on profile likelihood, and a Bayesian approach based on Markov-chain Monte Carlo. Through three case studies, we demonstrate our method by providing a didactic guide to inference and identifiability analysis of hyperparameters that relate to the statistical moments of model parameters from independent observed data. Our approach has a computational cost comparable to analysis of models that neglect heterogeneity, a significant improvement over many existing alternatives. We demonstrate how analysis of random parameter models can aid better understanding of the sources of heterogeneity from biological data.

Formation and Growth of Co-Culture Tumour Spheroids: New Compartment-Based Mathematical Models and Experiments.

Co-culture tumour spheroid experiments are routinely performed to investigate cancer progression and test anti-cancer therapies. Therefore, methods to quantitatively characterise and interpret co-culture spheroid growth are of great interest. However, co-culture spheroid growth is complex. Multiple biological processes occur on overlapping timescales and different cell types within the spheroid may have different characteristics, such as differing proliferation rates or responses to nutrient availability. At present there is no standard, widely-accepted mathematical model of such complex spatio-temporal growth processes. Typical approaches to analyse these experiments focus on the late-time temporal evolution of spheroid size and overlook early-time spheroid formation, spheroid structure and geometry. Here, using a range of ordinary differential equation-based mathematical models and parameter estimation, we interpret new co-culture experimental data. We provide new biological insights about spheroid formation, growth, and structure. As part of this analysis we connect Greenspan's seminal mathematical model to co-culture data for the first time. Furthermore, we generalise a class of compartment-based spheroid mathematical models that have previously been restricted to one population so they can be applied to multiple populations. As special cases of the general model, we explore multiple natural two population extensions to Greenspan's seminal model and reveal biological mechanisms that can describe the internal dynamics of growing co-culture spheroids and those that cannot. This mathematical and statistical modelling-based framework is well-suited to analyse spheroids grown with multiple different cell types and the new class of mathematical models provide opportunities for further mathematical and biological insights.

Reliable and efficient parameter estimation using approximate continuum limit descriptions of stochastic models.

Stochastic individual-based mathematical models are attractive for modelling biological phenomena because they naturally capture the stochasticity and variability that is often evident in biological data. Such models also allow us to track the motion of individuals within the population of interest. Unfortunately, capturing this microscopic detail means that simulation and parameter inference can become computationally expensive. One approach for overcoming this computational limitation is to coarse-grain the stochastic model to provide an approximate continuum model that can be solved using far less computational effort. However, coarse-grained continuum models can be biased or inaccurate, particularly for certain parameter regimes. In this work, we combine stochastic and continuum mathematical models in the context of lattice-based models of two-dimensional cell biology experiments by demonstrating how to simulate two commonly used experiments: cell proliferation assays and barrier assays. Our approach involves building a simple statistical model of the discrepancy between the expensive stochastic model and the associated computationally inexpensive coarse-grained continuum model. We form this statistical model based on a limited number of expensive stochastic model evaluations at design points sampled from a user-chosen distribution, corresponding to a computer experiment design problem. With straightforward design point selection schemes, we show that using the statistical model of the discrepancy in tandem with the computationally inexpensive continuum model allows us to carry out prediction and inference while correcting for biases and inaccuracies due to the continuum approximation. We demonstrate this approach by simulating a proliferation assay, where the continuum limit model is the well-known logistic ordinary differential equation, as well as a barrier assay where the continuum limit model is closely related to the well-known Fisher-KPP partial differential equation. We construct an approximate likelihood function for parameter inference, both with and without discrepancy correction terms. Using maximum likelihood estimation, we provide point estimates of the unknown parameters, and use the profile likelihood to characterise the uncertainty in these estimates and form approximate confidence intervals. For the range of inference problems considered, working with the continuum limit model alone leads to biased parameter estimation and confidence intervals with poor coverage. In contrast, incorporating correction terms arising from the statistical model of the model discrepancy allows us to recover the parameters accurately with minimal computational overhead. The main tradeoff is that the associated confidence intervals are typically broader, reflecting the additional uncertainty introduced by the approximation process. All algorithms required to replicate the results in this work are written in the open source Julia language and are available at GitHub.

Parameter identifiability and model selection for sigmoid population growth models.

Sigmoid growth models, such as the logistic, Gompertz and Richards' models, are widely used to study population dynamics ranging from microscopic populations of cancer cells, to continental-scale human populations. Fundamental questions about model selection and parameter estimation are critical if these models are to be used to make practical inferences. However, the question of parameter identifiability - whether a data set contains sufficient information to give unique or sufficiently precise parameter estimates - is often overlooked. We use a profile-likelihood approach to explore practical parameter identifiability using data describing the re-growth of hard coral. With this approach, we explore the relationship between parameter identifiability and model misspecification, finding that the logistic growth model does not suffer identifiability issues for the type of data we consider whereas the Gompertz and Richards' models encounter practical non-identifiability issues. This analysis of parameter identifiability and model selection is important because different growth models are in biological modelling without necessarily considering whether parameters are identifiable. Standard practices that do not consider parameter identifiability can lead to unreliable or imprecise parameter estimates and potentially misleading mechanistic interpretations. For example, using the Gompertz model, the estimate of the time scale of coral re-growth is 625 days when we estimate the initial density from the data, whereas it is 1429 days using a more standard approach where variability in the initial density is ignored. While tools developed here focus on three standard sigmoid growth models only, our theoretical developments are applicable to any sigmoid growth model and any appropriate data set. MATLAB implementations of all software are available on GitHub.

A Continuum Mathematical Model of Substrate-Mediated Tissue Growth.

We consider a continuum mathematical model of biological tissue formation inspired by recent experiments describing thin tissue growth in 3D-printed bioscaffolds. The continuum model, which we call the substrate model, involves a partial differential equation describing the density of tissue, [Formula: see text] that is coupled to the concentration of an immobile extracellular substrate, [Formula: see text]. Cell migration is modelled with a nonlinear diffusion term, where the diffusive flux is proportional to [Formula: see text], while a logistic growth term models cell proliferation. The extracellular substrate [Formula: see text] is produced by cells and undergoes linear decay. Preliminary numerical simulations show that this mathematical model is able to recapitulate key features of recent tissue growth experiments, including the formation of sharp fronts. To provide a deeper understanding of the model we analyse travelling wave solutions of the substrate model, showing that the model supports both sharp-fronted travelling wave solutions that move with a minimum wave speed, [Formula: see text], as well as smooth-fronted travelling wave solutions that move with a faster travelling wave speed, [Formula: see text]. We provide a geometric interpretation that explains the difference between smooth and sharp-fronted travelling wave solutions that is based on a slow manifold reduction of the desingularised three-dimensional phase space. In addition, we also develop and test a series of useful approximations that describe the shape of the travelling wave solutions in various limits. These approximations apply to both the sharp-fronted and smooth-fronted travelling wave solutions. Software to implement all calculations is available at GitHub .

Persistence as an Optimal Hedging Strategy.

Bacteria invest in a slow-growing subpopulation, called persisters, to ensure survival in the face of uncertainty. This hedging strategy is remarkably similar to financial hedging, where diversifying an investment portfolio protects against economic uncertainty. We provide a new, to our knowledge, theoretical foundation for understanding cellular hedging by unifying the study of biological population dynamics and the mathematics of financial risk management through optimal control theory. Motivated by the widely accepted role of volatility in the emergence of persistence, we consider several models of environmental volatility described by continuous-time stochastic processes. This allows us to study an emergent cellular hedging strategy that maximizes the expected per capita growth rate of the population. Analytical and simulation results probe the optimal persister strategy, revealing results that are consistent with experimental observations and suggest new opportunities for experimental investigation and design. Overall, we provide a new, to our knowledge, way of conceptualizing and modeling cellular decision making in volatile environments by explicitly unifying theory from mathematical biology and finance.

Synchronized oscillations in growing cell populations are explained by demographic noise.

Understanding synchrony in growing populations is important for applications as diverse as epidemiology and cancer treatment. Recent experiments employing fluorescent reporters in melanoma cell lines have uncovered growing subpopulations exhibiting sustained oscillations, with nearby cells appearing to synchronize their cycles. In this study, we demonstrate that the behavior observed is consistent with long-lasting transient phenomenon initiated and amplified by the finite-sample effects and demographic noise. We present a novel mathematical analysis of a multistage model of cell growth, which accurately reproduces the synchronized oscillations. As part of the analysis, we elucidate the transient and asymptotic phases of the dynamics and derive an analytical formula to quantify the effect of demographic noise in the appearance of the oscillations. The implications of these findings are broad, such as providing insight into experimental protocols that are used to study the growth of asynchronous populations and, in particular, those investigations relating to anticancer drug discovery.

The role of mechanical interactions in EMT.

The detachment of cells from the boundary of an epithelial tissue and the subsequent invasion of these cells into surrounding tissues is important for cancer development and wound healing, and is strongly associated with the epithelial-mesenchymal transition (EMT). Chemical signals, such as TGF-ß, produced by surrounding tissue can be uptaken by cells and induce EMT. In this work, we present a novel cell-based discrete mathematical model of mechanical cellular relaxation, cell proliferation, and cell detachment driven by chemically-dependent EMT in an epithelial tissue. A continuum description of the model is then derived in the form of a novel nonlinear free boundary problem. Using the discrete and continuum models we explore how the coupling of chemical transport and mechanical interactions influences EMT, and postulate how this could be used to help control EMT in pathological situations.

Modelling cell guidance and curvature control in evolving biological tissues.

Tissue geometry is an important influence on the evolution of many biological tissues. The local curvature of an evolving tissue induces tissue crowding or spreading, which leads to differential tissue growth rates, and to changes in cellular tension, which can influence cell behaviour. Here, we investigate how directed cell motion interacts with curvature control in evolving biological tissues. Directed cell motion is involved in the generation of angled tissue growth and anisotropic tissue material properties, such as tissue fibre orientation. We develop a new cell-based mathematical model of tissue growth that includes both curvature control and cell guidance mechanisms to investigate their interplay. The model is based on conservation principles applied to the density of tissue synthesising cells at or near the tissue's moving boundary. The resulting mathematical model is a partial differential equation for cell density on a moving boundary, which is solved numerically using a hybrid front-tracking method called the cell-based particle method. The inclusion of directed cell motion allows us to model new types of biological growth, where tangential cell motion is important for the evolution of the interface, or for the generation of anisotropic tissue properties. We illustrate such situations by applying the model to simulate both the resorption and infilling components of the bone remodelling process, and to simulate root hair growth. We also provide user-friendly MATLAB code to implement the algorithms.

Model-based data analysis of tissue growth in thin 3D printed scaffolds.

Tissue growth in three-dimensional (3D) printed scaffolds enables exploration and control of cell behaviour in more biologically realistic geometries than that allowed by traditional 2D cell culture. Cell proliferation and migration in these experiments have yet to be explicitly characterised, limiting the ability of experimentalists to determine the effects of various experimental conditions, such as scaffold geometry, on cell behaviour. We consider tissue growth by osteoblastic cells in melt electro-written scaffolds that comprise thin square pores with sizes that were deliberately increased between experiments. We collect highly detailed temporal measurements of the average cell density, tissue coverage, and tissue geometry. To quantify tissue growth in terms of the underlying cell proliferation and migration processes, we introduce and calibrate a mechanistic mathematical model based on the Porous-Fisher reaction-diffusion equation. Parameter estimates and uncertainty quantification through profile likelihood analysis reveal consistency in the rate of cell proliferation and steady-state cell density between pore sizes. This analysis also serves as an important model verification tool: while the use of reaction-diffusion models in biology is widespread, the appropriateness of these models to describe tissue growth in 3D scaffolds has yet to be explored. We find that the Porous-Fisher model is able to capture features relating to the cell density and tissue coverage, but is not able to capture geometric features relating to the circularity of the tissue interface. Our analysis identifies two distinct stages of tissue growth, suggests several areas for model refinement, and provides guidance for future experimental work that explores tissue growth in 3D printed scaffolds.

Biologically-informed neural networks guide mechanistic modeling from sparse experimental data.

Biologically-informed neural networks (BINNs), an extension of physics-informed neural networks [1], are introduced and used to discover the underlying dynamics of biological systems from sparse experimental data. In the present work, BINNs are trained in a supervised learning framework to approximate in vitro cell biology assay experiments while respecting a generalized form of the governing reaction-diffusion partial differential equation (PDE). By allowing the diffusion and reaction terms to be multilayer perceptrons (MLPs), the nonlinear forms of these terms can be learned while simultaneously converging to the solution of the governing PDE. Further, the trained MLPs are used to guide the selection of biologically interpretable mechanistic forms of the PDE terms which provides new insights into the biological and physical mechanisms that govern the dynamics of the observed system. The method is evaluated on sparse real-world data from wound healing assays with varying initial cell densities [2].

Invading and Receding Sharp-Fronted Travelling Waves.

Biological invasion, whereby populations of motile and proliferative individuals lead to moving fronts that invade vacant regions, is routinely studied using partial differential equation models based upon the classical Fisher-KPP equation. While the Fisher-KPP model and extensions have been successfully used to model a range of invasive phenomena, including ecological and cellular invasion, an often-overlooked limitation of the Fisher-KPP model is that it cannot be used to model biological recession where the spatial extent of the population decreases with time. In this work, we study the Fisher-Stefan model, which is a generalisation of the Fisher-KPP model obtained by reformulating the Fisher-KPP model as a moving boundary problem. The nondimensional Fisher-Stefan model involves just one parameter, [Formula: see text], which relates the shape of the density front at the moving boundary to the speed of the associated travelling wave, c. Using numerical simulation, phase plane and perturbation analysis, we construct approximate solutions of the Fisher-Stefan model for both slowly invading and receding travelling waves, as well as for rapidly receding travelling waves. These approximations allow us to determine the relationship between c and [Formula: see text] so that commonly reported experimental estimates of c can be used to provide estimates of the unknown parameter [Formula: see text]. Interestingly, when we reinterpret the Fisher-KPP model as a moving boundary problem, many overlooked features of the classical Fisher-KPP phase plane take on a new interpretation since travelling waves solutions with [Formula: see text] are normally disregarded. This means that our analysis of the Fisher-Stefan model has both practical value and an inherent mathematical value.

Mathematical Model of Tumour Spheroid Experiments with Real-Time Cell Cycle Imaging.

Three-dimensional (3D) in vitro tumour spheroid experiments are an important tool for studying cancer progression and potential cancer drug therapies. Standard experiments involve growing and imaging spheroids to explore how different conditions lead to different rates of spheroid growth. These kinds of experiments, however, do not reveal any information about the spatial distribution of the cell cycle within the expanding spheroid. Since 2008, a new experimental technology called fluorescent ubiquitination-based cell cycle indicator (FUCCI) has enabled real-time in situ visualisation of the cell cycle progression. Observations of 3D tumour spheroids with FUCCI labelling reveal significant intratumoural structure, as the cell cycle status can vary with location. Although many mathematical models of tumour spheroid growth have been developed, none of the existing mathematical models are designed to interpret experimental observations with FUCCI labelling. In this work, we adapt the mathematical framework originally proposed by Ward and King (Math Med Biol 14:39-69, 1997. https://doi.org/10.1093/imammb/14.1.39 ) to produce a new mathematical model of FUCCI-labelled tumour spheroid growth. The mathematical model treats the spheroid as being composed of three subpopulations: (i) living cells in G1 phase that fluoresce red; (ii) living cells in S/G2/M phase that fluoresce green; and (iii) dead cells that are not fluorescent. We assume that the rates at which cells pass through different phases of the cell cycle, and the rate of cell death, depend upon the local oxygen concentration. Parameterising the new mathematical model using experimental measurements of cell cycle transition times, we show that the model can qualitatively capture important experimental observations that cannot be addressed using previous mathematical models. Further, we show that the mathematical model can be used to qualitatively mimic the action of anti-mitotic drugs applied to the spheroid. All software programs required to solve the nonlinear moving boundary problem associated with the new mathematical model are available on GitHub. at https://github.com/wang-jin-mathbio/Jin2021.

Extinction of Bistable Populations is Affected by the Shape of their Initial Spatial Distribution.

The question of whether biological populations survive or are eventually driven to extinction has long been examined using mathematical models. In this work, we study population survival or extinction using a stochastic, discrete lattice-based random walk model where individuals undergo movement, birth and death events. The discrete model is defined on a two-dimensional hexagonal lattice with periodic boundary conditions. A key feature of the discrete model is that crowding effects are introduced by specifying two different crowding functions that govern how local agent density influences movement events and birth/death events. The continuum limit description of the discrete model is a nonlinear reaction-diffusion equation, and we focus on crowding functions that lead to linear diffusion and a bistable source term that is often associated with the strong Allee effect. Using both the discrete and continuum modelling tools, we explore the complicated relationship between the long-term survival or extinction of the population and the initial spatial arrangement of the population. In particular, we study different spatial arrangements of initial distributions: (i) a well-mixed initial distribution where the initial density is independent of position in the domain; (ii) a vertical strip initial distribution where the initial density is independent of vertical position in the domain; and, (iii) several forms of two-dimensional initial distributions where the initial population is distributed in regions with different shapes. Our results indicate that the shape of the initial spatial distribution of the population affects extinction of bistable populations. All software required to solve the discrete and continuum models used in this work are available on GitHub .