Preparation:

Step 1: Prepare the Crust

1. Preheat your oven to 325°F (163°C) and line a muffin tin with cupcake liners.
2. In a mixing bowl, combine the graham cracker crumbs, sugar, melted butter, and lime zest. Mix until the crumbs are evenly coated.
3. Divide the mixture evenly among the cupcake liners, pressing down firmly to create a compact crust.
4. Bake for 5-6 minutes, then remove from the oven and let cool while preparing the filling.

Step 2: Make the Blackberry Swirl

1. In a small saucepan over medium heat, combine the blackberries, sugar, cornstarch, lime juice, and water.
2. Cook for about 5-7 minutes, stirring and mashing the berries until the mixture thickens into a sauce-like consistency.
3. Strain the blackberry mixture through a fine-mesh sieve to remove seeds, then set aside to cool.

Step 3: Prepare the Cheesecake Filling

1. In a large mixing bowl, beat the softened cream cheese and sugar until smooth and creamy.
2. Add eggs one at a time, mixing on low speed until just combined.
3. Stir in the vanilla extract, lime juice, lime zest, sour cream, and flour, mixing until smooth.

Step 4: Assemble the Cheesecake Cupcakes

1. Fill each cupcake liner with cheesecake batter, filling about 3/4 full.
2. Drop small dollops of the blackberry swirl mixture on top of each cheesecake.
3. Use a toothpick or skewer to gently swirl the blackberry sauce into the batter for a marbled effect.

Step 5: Bake and Cool

1. Bake the cheesecakes for 18-22 minutes, or until the centers are set but slightly jiggly.
2. Turn off the oven, crack the door open slightly, and let the cheesecakes cool inside for 10 minutes.
3. Transfer the muffin tin to a wire rack to cool completely, then refrigerate for at least 3-4 hours (preferably overnight) before serving.