Understanding Alien Species Co-Invasions: Mathematical Models & Analysis

Introduction

This educational webpage explores the mathematical modeling of alien species co-invasions based on the paper "Mixed additive modeling of global alien species co-invasions." The paper presents a sophisticated statistical framework for understanding how plants and insects co-invade different regions around the world.

The key innovation in the paper is the use of a mixed additive Relational Event Model (REM) that can handle:

Time-varying effects (how factors like trade impact invasions differently over time)
Random effects (accounting for differences between species and regions)
Co-invasion patterns (how the presence of one species affects invasions by others)

This webpage will help you understand the mathematical concepts, inference methods, and simulation techniques used in the paper. We'll also extend the analysis to fungi species, providing a foundation for further research in this area.

Cumulative number of first records over time

Figure 1: Cumulative number of first records of alien species over time, showing the acceleration of invasions.

Mathematical Framework

Relational Event Model (REM)

The paper models alien species invasions as a marked point process, where each "event" represents the first record of a particular species in a specific region. This approach allows us to model the timing and patterns of invasions.

The fundamental concept is the hazard function, which represents the instantaneous probability of an invasion event occurring at time \(t\): \[ \lambda(t; \theta) = \lim_{\Delta t \to 0} \frac{P(t \leq T < t + \Delta t | T \geq t)}{\Delta t} \] (1) Where \(T\) is the random variable representing the time of invasion, and \(\theta\) represents the model parameters.

Mixed Additive Model

The paper uses a mixed additive model to incorporate various factors affecting invasion rates:

\[ \lambda_{ij}(t) = \exp\left( \alpha + \sum_{k=1}^{K} \beta_k x_{ijk}(t) + \sum_{l=1}^{L} f_l(z_{ijl}(t)) + u_i + v_j \right) \] (2) Where: \(\lambda_{ij}(t)\) is the hazard rate for species \(i\) invading region \(j\) at time \(t\) \(\alpha\) is the baseline hazard rate \(\beta_k\) are coefficients for linear effects \(x_{ijk}(t)\) \(f_l(z_{ijl}(t))\) are smooth functions of covariates \(z_{ijl}(t)\) \(u_i\) and \(v_j\) are random effects for species \(i\) and region \(j\)

Time-Varying Effects

One of the key innovations in the paper is modeling how the effects of covariates change over time:

\[ \beta_k(t) = \sum_{m=1}^{M} \gamma_{km} B_m(t) \] (3) Where \(B_m(t)\) are basis functions (such as B-splines) and \(\gamma_{km}\) are coefficients to be estimated.

Co-Invasion Effects

The paper models how the presence of one species affects the invasion probability of another:

\[ x_{ijk}^{\text{co-invasion}}(t) = \sum_{i' \in S_{i}} \mathbb{1}(T_{i'j} < t) \] (4) Where \(S_{i}\) is the set of species related to species \(i\), and \(\mathbb{1}(T_{i'j} < t)\) indicates whether species \(i'\) has already invaded region \(j\) before time \(t\).

Figure 2: Number of first records by decade, showing temporal patterns in invasion rates.

Inference Methods

Case-Control Sampling

Estimating the full model with all possible species-region-time combinations would be computationally infeasible. The paper uses case-control sampling to make the estimation tractable:

The log-likelihood for the case-control sample is: \[ \ell(\theta) = \sum_{(i,j,t) \in \mathcal{D}} \log \lambda_{ij}(t) - \sum_{(i,j,t) \in \mathcal{D} \cup \mathcal{C}} \log(1 + \lambda_{ij}(t)) \] (5) Where \(\mathcal{D}\) is the set of observed invasions (cases) and \(\mathcal{C}\) is the set of sampled non-invasions (controls).

Penalized Maximum Likelihood

To estimate the smooth functions and prevent overfitting, the paper uses penalized maximum likelihood:

\[ \ell_p(\theta) = \ell(\theta) - \frac{1}{2} \sum_{l=1}^{L} \lambda_l \int [f_l''(z)]^2 dz \] (6) Where \(\lambda_l\) are smoothing parameters that control the trade-off between fit and smoothness.

Goodness-of-Fit Evaluation

The paper evaluates model fit using several metrics:

AIC (Akaike Information Criterion): Balances model fit and complexity
Temporal Residuals: Assess if the model captures temporal patterns correctly
ROC Curves: Evaluate the model's predictive performance

Figure 3: Top regions by number of first records, showing geographic patterns in invasions.

Data Analysis

Global Alien Species First Record Database

The analysis uses the Global Alien Species First Record Database, which contains records of when alien species were first detected in different regions around the world.

Key Database Statistics:

Total records: 61,751
Date range: Ancient times to 2020
Plants (Tracheophyta): 32,098 records (52%)
Insects (Insecta): 10,801 records (17.5%)
Fungi: 681 records (1.1%)

Temporal Patterns

The analysis reveals clear temporal patterns in invasion rates:

Invasion rates have accelerated dramatically over time
Different taxonomic groups show different temporal patterns
There are notable peaks in invasion rates corresponding to historical events

Heatmap of invasions by decade and region

Figure 4: Heatmap showing invasion patterns across regions and decades.

Geographic Patterns

The analysis also reveals geographic patterns in invasions:

Some regions (e.g., United States, Australia) have experienced more invasions than others
Island regions are particularly vulnerable to invasions
Geographic patterns differ between taxonomic groups

Co-Invasion Patterns

The analysis examines co-invasion patterns between plants and insects:

There is a positive correlation between plant and insect invasions
This correlation suggests potential facilitation effects
The strength of correlation varies across regions and time periods

Figure 5: Top species by number of regions invaded, showing the most widespread invasive species.

Extension to Fungi Species

Fungi in the Database

The Global Alien Species First Record Database contains 681 fungi records (1.1% of all records), primarily from two phyla:

Ascomycota: 434 records (63.7%)
Basidiomycota: 238 records (35.0%)

Figure 6: Correlation between fungi and plant invasions, showing potential ecological relationships.

Fungi Invasion Patterns

The analysis of fungi invasions reveals several interesting patterns:

Moderate correlation between fungi and plant invasions (0.359)
Weaker correlation between fungi and insect invasions (0.182)
Temporal patterns differ from those of plants and insects
Geographic distribution shows preferences for certain regions

Figure 7: Top fungi species by number of regions invaded.

Adapting the Mathematical Framework to Fungi

To extend the mixed additive REM framework to fungi invasions, we need to consider several factors:

1. Specific Covariates for Fungi

Substrate availability (wood, soil, plant material)
Humidity and moisture levels
Host plant presence (for symbiotic or parasitic fungi)
Forest cover and composition
Soil pH and composition

2. Random Effects Structure

Species-specific dispersal mechanisms (spore type, size)
Functional traits (saprophytic, mycorrhizal, parasitic)
Region-specific factors (forest management practices)

3. Co-invasion Dynamics

Plant-fungi interactions (mycorrhizal relationships, plant pathogens)
Insect-fungi interactions (insect vectors, entomopathogenic fungi)
Facilitation effects (how presence of certain species affects fungi invasion)

4. Time-varying Effects

Changes in forestry and agricultural practices
Climate change impacts on fungi distribution
Changes in international trade of wood, plant, and soil products

5. Data Challenges

Fungi are often underreported compared to plants and animals
Taxonomic uncertainties and cryptic species
Detection biases (fungi are often only detected when fruiting bodies appear)

Figure 8: Correlation between fungi and insect invasions.

Interactive Simulations

Hazard Function Simulation

This simulation demonstrates how the hazard function changes with different parameter values. The hazard function represents the instantaneous probability of an invasion event occurring at a given time.